Cylindric Set Algebras

Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann

883 Cylindric Set Algebras

Cylindric Set Algebras and Related Structures By L. Henkin, J. D. Monk, and A. Tarski On Cylindric-Relativized Set Algebras By H. Andreka and I. Nemeti

Springer-Verlag Berlin Heidelberg New York 1981

Authors

Leon Henkin Department of Mathematics, University of California Berkeley, CA 94720, USA J. Donald Monk Department of Mathematics, University of Colorado Boulder, CO 80309, USA Alfred Tarski 462 Michigan Ave. Berkeley, CA 94707, USA Hajnalka Andreka Istvan Nemeti Mathematical Institute, Hungarian Academy of Sciences Realtanoda u. 13-15, 1053 Budapest, Hungary

AMS Subject Classifications (1980): 03 C55, 03 G 15

ISBN 3-540-10881-5 Springer-Verlag Berlin Heidelberg NewYork ISBN 0-387-10881-5 Springer-Verlag NewYork Heidelberg Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of iltustTations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under w 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort", Munich. 9 by Springer-Verlag Berlin Heidelberg 1981 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210

Introduction

This volume theoretical

is devoted to a comprehensive

treatment of certain set-

structures which consist of fields of sets enhanced by addi-

tional fundamental

operations and distinguished

elements.

The treatment

dimension

~

is largely self-contained. Each of these structures has an associated infinite ordinal; Let 3

R

R

their basic form is well illustrated

be an arbitrary

set, and let

of all triples of elements of

of subsets of relative to tions

3R

~

R .

Thus

~

~ = 3 .

is a non-empty collection

intersection,

We shall assume that

CO, Cl, C 2

in the case

be a field of subsets of the set

closed under union,

3R .

, a finite or

~

of cylindriflcation,

and complementatlen

is closed under the three opera-

where

C 0 , for example,

is the

operation given by:

CoX = {<x,y,z> : (u,y,z> ~ X

C0X is the cylinder formed by moving and

C2X

are similarly

t h a t the diagonal planes

for some

X

parallel

D02

(resp.,

and third (resp.,

D01, D02, D12

DI2 )

a r e in

3).

coincide.

is called a cylindric

3R

whose first

A collection field of sets

fields of sets and certain closely related

structures are the objects of study in this volume. as collections

We also assume

.

second and third) coordinates

Cylindric

CIX

~ ; h e r e , for example,

consists of all triples of

satisfying all of these conditions (of dimension

to the first axis.

related to the second and third axes.

D01={<x,x,y>: x , y ~ R } Similarly

u, with xER};

of sets, hut as algebraic

Considered not merely

objects endowed with fundamental

iV

Operations and distinguished elements, cylindric fields of sets are called cylindric set algebras. of dimension

~ , and

Cs ICs

is the class of all cylindric set algebras is the class of algebras isomorphic to them.

In much of the work, general algebraic notions are studied in their application to cylindric set algebras.

We consider subalgebras, homomorphisms,

products, and ultraproducts of them, paying special attention, for example, to the closure of

ICs

and related classes under these operations.

In

addition, there are natural operations upon these structures which are specific to their form as certain Boolean algebras with operators, such as relativization to subsets of 3S

with

3R

and isomorphism to algebras of subsets of

S ~ R , and there are relationships between set algebras of

different dimensions. Although, as mentioned, the volume is largely self-contained, we shall often refer to the book Cylindric Algebras~ Part I, by Henkin, Monk, and Tarski.

Many notions touched on briefly in the present volume are treated

in detail in that one, and motivation for considering certain questions can be found there.

Indeed, the present work had its genesis in the decision

by Henkin, Monk, and Tarski to publish a series of papers which would form the bulk of Part II of their earlier work.

Their contribution to the present

volume is, in fact, the first of this proposed series.

As their writing l

proceeded, they learned of the closely related results obtained by Andreka and Nimeti, and invited the latter to publish jointly with themselves. Thus, the present volume consists of two parts.

The first, by Henkln,

Monk, and Tarskl, contains the basic defintions and results on various kinds of cylindric set algebras. parallel to the first.

The second, by Andrlka and Nemeti, is organized In it, certain aspects of the theory are investigated

more thoroughly; in particular, many results which are merely formulated

V

in Part I, are provided with proofs in Part II.

In both parts, many

open problems concerning the structures considered are presented.

The authors Berkeley Boulder Budapest

Table

First

Part: L.

Cylindric set algebras Henkin, J.D~ Monk, and set

and related structures, by A. T a r s k i ..................

2.

Relativization

....................................

12

3.

Change

....................................

33

4.

Subalgebras

.......................................

56

5.

Homomorphisms

.....................................

66

6.

Products

7.

Ultraproducts

8.

Reducts

9.

Problems

base

..............................

i

Various

of

algebras

contents

1.

..........................................

References

Second

of

.....................................

4

73 86

...........................................

122

..........................................

126

............................................

128

Part: On cylindric-relativized set algebras, by H Andr~ka a n d I. N ~ m e t i .............................. concepts

O

Basic

i

Regular

2

Relativization

.....................................

153

3

Change

.....................................

155

4

Subalgebras

........................................

185

5

Homomorphisms

......................................

209

6

Products

7

Ultraproducts

8

Reducts

9

Problems

References

Index

of

symbols

Index

of

defined

and

cylindric

of

base

notations

131

set

algebras

.......................

132

.....................

145

........................................... ......................................

222 229

............................................

261

...........................................

310

............................................

314

...............................................

317

terms

.........................................

322

Cylindric

set algebras and related structures

by L. Henkln,

The abstract the book [HMT] mentioned,

J.D. Monk, and A. Tarskl

theory of cylindric algebras

by the authors.

primarily

is extensively

Several kinds of special

for motivational

purpose of this article

I)

purposes,

developed

set algebras were

in that book.

It is the

to begin the examination of these set algebras

more detail.

The simplest and most important kind of set algebras are

the cylindric

set algebras

introduced

we shall refer to items from

in 1.1.5.

(Throughout

Recall that the unit element of any

set algebra

(Cs)

is the Cartesian power

and the other elements of DKk

of

~

is

the set

A

and for each

of cyllndrlflcatlon

are subsets of

ix E ~U: x K = x k }

fundamental Boolean operations mentatlon;

K < ~

of

~

~U

of a set

~U .

K,h < ~

(the base),

"

are union, intersection,

the fundamental operation

their definition is similar to that of a cylindric (cf. 2.2.11),

U

cylindric

The diagonal element

for each

by translation parallel to the

set algebras

this article

~ - dimensional

CK

the

and compleconsists

Kth axis of the space.

Several other kinds of set algebras were briefly considered

cylindric

in [HMT], and

set algebra:

generalized cylindric

weak

set algebras

(of. 1.1.13), and what we shall now call generalized weak cylindric algebras

(cf. 2.2.11).

unit elements

in

[HMT] by number without explicitly mentioning

that book).

~

in

The algebras

set

of each of these kinds have for their

subsets of a special kind of some Cartesian

space

~U , while

I) This article is the first in a series intended to form a large portion of the second volume of the work Cylindric Algebras, of which Part I has appeared ([HMT] in the bibliography). The research and writing were supported in part by NSF grants MPS 75-03583j MCS 77-22913.

the fundamental a

Cs

operations

of any such algebra are obtained

, with unit element

to the unit element

~U

of the algebra discussed.

these several classes

of cylindric

general class of set algebras algebras, space.

, by relativization

set algebras

in any detail,

relativization w h i c h are

directly relevant

first-order

discourse

ourselves to the aspects

to our d i s c u s s i o n

of

of those set algebras

A

forms an

for

~

free

regularity:

if (Here .)

~

Furthermore,

x E B ~gx

Regular

,

f E x

,

, the dimension

of

of [HMT]).

and any

A

field of sets, (for the

Thus the above

set algebra i. Ii.i

K < ~ ~

~

cylindric

of variables

in the sense of

for the finitely many

in

structure

~ - dimensional

of a cylindric

is locally finite dimensional

stems from the

, the collection

see the Preliminaries

is the universe

except possibly

set algebras

Given any relational

is the length of the sequence

collection

1.6.1

set algebras.

the class of all cylindric-relativized

of cylindric

language

~ a formula of A]

notation used here,

see

that are

CA's

following construction.

g E x

of

set

of those algebras

from full cylindric

restricting

Much of the importance

occurs

2.2)

in w h i c h the unit elements may be arbitrary subsets of a Cartesian

We shall not discuss here, however,

~

To unify our treatment

that of the cylindric-relativized

These algebras are simply subalgebras

where

(in the sense of

set algebras we use here as the most

obtained by arbitrary relativizations

[~:

from those of

~

.

This algebra

, since

such that the

CK~ ~ Kth

variable

has an additional property of

g E

C

set of

, and x

, is

Ax~f = ~xlg

, then

~K E ~ : cKx f x}

set algebras will be discussed e x t e n s i v e l y

The article has nine sections.

=

later.

In section 1 we give formal defini-

tions of the classes of set algebras w h i c h are studied in this article and we state the simplest relationships found in later sections of the paper. ships are established,

b e t w e e n them; the proofs are In section 2 some deeper relation-

using the n o t i o n of relativization.

is concerned with change of base,

treating

Section 3

the question of conditions

;

of

under which a set algebra with base different base L~wenheim,

W

U

; the main results are algebraic versions

Skolem, Tarski theorems

are also found in section 7). subalgebra is investigated cular attention

is isomorphic to one with a of the

(some results on change of base

In section 4 the algebraic notion of

for our various

set algebras,

paying parti-

to the problem about the minimum number of generators

for a set algebra.

Homomorphisms

of set algebras are discussed

tion 5, and products, along with the related indecomposability are studied in their application 7, devoted to ultraproducts results in the paper.

of set algebras,

In particular,

less trivial of the relationships described

to set algebras

in section 6.

gives perhaps

in secnotions, Section

the deepest

it is in this section that the

between the classes of set algebras

in section 1 are established.

set algebras are discussed in section 8.

Reducts and neat embeddings of Finally,

in section 9 we list

the most important problems concerning set algebras which are open at this time, and we also take this opportunity of the problems

stated in

For reference

[HMT]

to describe the status

.

in later articles, we refer to theorems, definitions,

etc., by three figures, e.g. 1.2.2 for the second item in section 2 of paper number I, which is just the present paper

(see the initial footnote).

The very most basic results on set algebras were first described in the paper Henkin, Tarski [HT]. in Henkin, Monk

[HM]

.

Other major results were obtained

In preparing

the present comprehensive

sion of set algebras many natural questions arose.

discus-

Some of these ques-

tions were solved by the authors, and their solutions are found here. A large number of the questions were solved by H. Andr~ka and I.N~meti. Where their solutions were short we have usually included the results here, with their permission, theirs.

and we have indicated

that the results are

Many of their longer solutions will be found in the paper

[AN3]

4

Various

following

set algebras

this one, which is organized

I.i.i

parallel to our paper; a few

of their related results are found in

JAN2],

course of our article we shall have occasion of their related results.

JAN4]

, or [N]

.

In the

to mention explicitly most

We are indebted to A n d r ~ k a and N~meti for

their considerable help in preparing this paper for publication, The following set-theoretical If

f E ~U

that

,

~ < ~ , and

(fK)>~ = fk u

if

u E U , then

k # < , while

reasons we sometimes write

1.

D e f i n i t i o n I.I.I. V ~ ~U

.

For all

notation not in [HMT] fK u

is

the

(fK)K = u . u

f(K/u)

in place of

will be useful.

member of

~U

such

For typographical fKu

y~arious set al$ebr~as

(i)

K,~ < ~

Let

U

be a set,

~

an ordinal,

and

we set

D[V] = [y E V : YK = Yk}

and we let

every

C IV]

be the mapping

SbV

into

SbV

such that,

V

u E U]

is implicitly understood we shall write

simply

A

= [yEV

K

for some

(ii)

: Yu E X

- dimensional

is an

iff there is a set

and a set

U

D

cylindric-relativized

V ~ ~U

such that

A

V

and

K < ~ ), and containing as elements

C IV] K

(for each

closed under all the operations

(for all

<,~ < ~ ).

(iii)

is a c y l i n d r i c - r e l a t i v i z e d

with base

~ U

The base of

iff there is a set

V ~ ~U

A

is the set

or

C

field of sets

U, N,

V

the subsets

UxEvRgx

.

set al~ebra of dimension such that

.)

is a non-empty

family of subsets of

D [V]

for

X -~ V ,

C[V]x (When

from

C~

I.l.l

Various

set algebras

=
' where

is the class of all c y l i n d r i c - r e l a t i v i z e d In case

relativized

(iv)

A = SbV

, the set

A

p E ~U

U

be a set and

space with base

U

V

and base

set algebras

and the algebra

~

U .

of dimension

are called res-

field of sets and a full cylindric-

~

an ordinal.

and dimension

~U

is then called the

~

Moreover,

for every

we set

~u (p) = {x E ~ U

and we call

~U (p)

determined b y

: {~<~

(v)

(v) - (ix)

, respectively

field of sets, all cylindric (vi) cylindric p ~ ~U

field of sets,

A

the form ~ distinct

iEl

~y

i,j ~ I . Gs

~

~

~

and dimension

Yi

that

is a c y l i n d r i c - r e l a t i v i z e d

, and unit element ~ - dimensional

V = ~U

.

V

cylindric

The class of

is denoted by

Cs

~ - dimensional weak

set algebra,

if there is a ~ - dimensional weak

Ws

, is called an

respectively

Yi ~ 0

The sets

U

The class of all

is denoted by

where

if

~

, is called an

respectively

field of sets, i '

, base

of dimension

, respectively

lized cylindric

U

of this d e f i n i t i o n we assume

, is called an

V = ~U (p) .

set algebras

(vii)

~

set algebra,

, respectively

such that

cylindric

~

respectively set algebras

A

is finite}

field of sets and

both with dimension

A

~ p~}

p

is a c y l i n d r i c - r e l a t i v i z e d

set algebra,

: x

the w e a k Cartesian space with base

For the following parts

symbol

~ - dimensional

set algebra.

Let

Cartesian

is an

field of sets with unit element

pectively a full c y l i n d r i c - r e l a t i v i z e d

A

A

for each

~-dimensional

set algebra, i E I

'

if

and

V

has

Y. N Y . = 0 l J

are called the subbases

denotes the class of all generalized

genera-

cylindric

of

~

.

for any two The

set algebras

6

Various

of dimension (viii)

A

weak cylindric
'

, respectively field of sets,

where

pi E ~Yi

for any two distinct .

We use

algebras

g E x

Gws

K

~ = 0

, is called an

respectively for each

x E A

x ~ A

in

Crs

all regular algebras w h i c h belong

~Lb~

Yi

V

~y~pi)

are called

~

generalized

has the form ~u -j

0

the subbases of

~

be a cylindric

formal definition). E K

K ,

Remark 1.1.2.

A

and

~

, are called regular.) K reg

the class of

K

algebra and

The d e f i n i t i o n

b E A ~

to

I.I.5. in

of cylindric

We denote by b

(see

2.2.1

in the introduction.

set algebras

Other parts of 1.1.13

and

regularity was defined only for cylindric

for a

I.I.i

2.2.11

in

l.l.l(v)

are consistent

o

Originally,

set algebras,

The general definition

in

in the form given

I.I.i

(ix)

is due to

H. Andr~ka and I. N~meti and turns out to have the desired meaning set algebras

as well as

Ws's

and

The inclusions holding among the various

consider

in

I.i.i

the classes

~ > 0 ;

b ~ A}

with the informal definitions

introduced

and

(We assume here

, we denote by

to

A

being a class of cylindric algebras, we let

coincides with d e f i n i t i o n

cylindric

set

that

is regular.

the algebra obtained by relativizing

R~K = [ ~ b ~ : N

if

(gxU 1)I f = (~xU i)1 g

as well as

being any class included

Let

and

is regular provided

f C x , g C V , and

, all elements,

(x)

set algebra,

i E I

The sets

~ - dimensional

~.

are called regular if each if

1.1.2

for the class of all generalized weak cylindric

An element

whenever

~

i,j E I .

of dimension

(ix)

set algebras

~ .

are indicated of isomorphic

then the diagram collapses

in Figure

's

; see

1.1.13 -1.1.16.

classes of set algebras 1.1.3

for

images of the various

as indicated

in each case are proper inclusions.

Gs

for

in Figure 1.1.4

~ > 0

; if we

set algebras, .

The inclusions

1.1.3

Various set algebras

7

C i s ~ .

crsreg

Gws ic

Gwsreg

i

~

~

sa,~

~ ~ G!reg

C s ~ .

Ws ~reg = W s

C reg 0~

Figure 1.1.3

ICrs I ~ C ~

eg

= I s

IGws

iCrs reg = IGws reg

ICs

r

iCs reg IWs

= IWs reg

Figure 1.1.4

In case

~ < w , the classes

and

; furthermore, under this assumption each member of any of

Gs

these classes is regular. to subdirect products of Gws's

and

Ws's

phic to a regular

. Gs

Ws

and

Cs

coincide, and so do

In the general case, Cs's

Every

Ws

Gs's

Gws

are isomorphic

and conversely; similarly for is regular.

Every

Gs

is isomor-

and to a subdireet product of regular

Cs 's.

Proofs of these facts and the relationships in the diagrams will be found at the appropriate places in this paper. We begin our discussion by describing some degenerate cases of the

8

Various

notions which

in

I.i.I

follow

, and giving

easily

omit proofs which

Corollary element

V

1.1.5.

Let

N

V = 0

iff

be a

(iii)

If

of > 0

and

U ~ 0 , or

Corollar~ ofU (p) = ~U

.

Cs

the paper we

w i t h base

U

and unit

Hence

Crs I

for

1.1.7.

V = 0 .

frequently

~ < ~

,

U

make

such a s s u m p t i o n s

is any set, and

~ < 0~ we have } , where

If

~

Gs

= Gws

p E ~d

, if

~of is the unique

as

, then

0 < ~ < w

Cs~

with universe

, where

the base

2

Gws

w i t h every

subbase

I ;

~I

and

Corollarx. 1.1.9.

Crs 0

(i)

(ii)

For

of ~ I ,

Crs

(iii)

For

of ~ 2 ,

Crs

N2

are the unique Furthermore,

is

0 .

For any

,

_c CA of

, where

Crs 0 = Gws 0 = Gs 0 = [~i,~2]

respectively.

of any

having

Gs

Crs I = Gws I = Gs I = Cs I = Ws I U [~}

with universe

and

is a

is a discrete

then

1.1.8.

Cs 0 = Ws 0 = [~2} I

= 2 .

Cs 0 = Ws 0 .

only one element,

Corollary

IAI

.

U = 0 , then

If

= Ws~ U [

Corollary

then

V ~ 0 .

1.1.6.

I , and finally

full, and

the classes

Throughout

V = [0}

V ~ ~0}

of this theorem we w i l l

universes

Crs

IAI = I ; if

of = 0 , then

we have

between

seem trivial.

If

of > 0 ,

inclusions

from the definitions.

(ii)

is the

those

1.1.5

.

(i)

Because

set algebras

~ CA

of of

Cs

U Gs

Crs0's

every

of

U Ws

Crs 0

~

U Gws

and with is

c CA

I.l.lO

Various Proof.

Both

we construct

~ E Crs

IuI > 1 , taking w i t h unit

cerning tion,

l9

Crs

trivial. by c h o o s i n g

, and letting We have

0 .

Thus

~

role

(iii),

U

with

be the full

D IV] = 0

Crs

so that

V =

axiom

(C6)

results

con-

'

to satisfy

,

is complete

w h y we

is introduced

shall not give many

Just

as indicated

to unify

in the introduc-

some definitions

and

results,

in our discussion.

If

I.I.I0.

any set

~

fails

in these papers;

an a u x i l i a r y

To e s t a b l i s h

~ ~ 2 , then

Cs

~ Gs

and

R~Cs

~ Crs

c~

We shall Gws

ffz2

9

O1

explains

this class

are

(l.l.l(iii)).

and the proof

Corollary SR%Cs

for

Ivl I0 ) =

the class

and plays

(ii)

~CA

V

'

~ ~ CA R

Corollary

and

v = ~u ~ -01 n [~U]

element

Ivl whence

(i)

set algebras

be able

c RgCs ~

although Rs [HR]

for

P~Cs O

c Crs

~ a 2 =

for

and Prop.2.3

Corollar~f C Gs

to strengthen

these

below by showing

(see 1 92 . 12 - 1.2,13).

Crs O

mnd

~ > 1

RgCs I

but we

=

Cs I

that

It is known

,

that

we have

shall not give an example

here"

see

and

Cs

of [AN3] (p.155),

I.I.II.

If

~ ~ w

, then

Ws

~ Gws

C Crs

~ Gws

Theorem

I 1.12.

Let

~

be a

9

where

results

Then

for all for any X

with

unit element

~

~. (pi) N ~ U (pj) = 0 ui j

~U! pi) E A l

Gws

E A

the

Also

assume

following

~u(pi)

~iEl

for all distinct

i E I .

~

i,j E I .

Assume

-i

that

~ ~ i . conditions

are equivalent:

'

10

Various (i)

X = ~u~i(pi)

(ii) and

&X

=

X 0

i E I ; (under

=

) such that

X ~ 0

.

(i) =

(ii)

~ U ~ pi) = 0 .

y E Y

for some

is a minimal element of

Proof. and

1.1.13

set algebras

and let

Clearly for any

Now suppose

x 6 ~, (pi) ui

,

such that

.

(~F)?y

that

.

we have

0 ~ Y ~ ~.(pi) ui

be arbitrary.

= (~F)~x

i E I

~us

There

and

0 ~ ~U~ pi) Ay = 0 .

is then a finite

x E c(F)Y = Y, and hence

Fix F ~

Y = ~u!Pi~ l

(ii) = # 0 . by

(i)

Assume

Since &~U! pi) = 0

(ii)

ui

established,

,

Gs

such that

X n ~U~ pi)

by 1.6.6

while by the implication

(i) =

Hence (ii)

already

,

~,u i(pi) = X N ~.u i(pi)

Now we discuss regular

iEl

(pi) &(X ~ U i )= 0

we have

X = X n ~ (pi) ,

Ws

(ii), and choose

Thus

set algebras.

(i)

holds.

In the case of the classes

the definition assumes a simpler form mentioned

Cs

in the

introduction.

Corollary I.I.13. X E A

.

~

be a

Cs

with base

U , and let

Then the following conditions are equivalent:

(i)

X is regular;

(ii) gEX

Let

for all

f E X

and all

(i) .

Trivial.

g E ~U , if

~XI f = AXJ g

then

.

Pro0f.

(ii) =

the hypotheses is obvious.

of

Suppose

(ii)

.

If

therefore

(&X ~, U I )~ fOgO = ( ~ X U I ) I g

The next two Corollaries

.

(i) =

(ii) .

Assume

(i)

0 E AX , the desired conclusion 0 ~X.

Hence

g E X

Then by

f0 gO E X

since

0 ~X

(i) .

are proved in the same way as

1.1.13.

and g E X and

1.1.14

Various

Corollary and

let

X E A

(i)

X

(ii) g E X

1.1.14. .

Let

Then

~

be a

the following

11

Ws

with

conditions

f ~ ~U ~p~

unit element are equivalent:

is regular; for all

f E X

and all

g ! ~U (p)

, if

AXI f = &X1g

then

.

Corollary where

Yi N y j

conditions

1.1.15.

= 0

Let

for

~I

i # j

be a

, and

Gs~

let

w i t h unit element

X E A .

Then the

~ i E l Y'l '

following

are equivalent:

(i)

X

(ii)

is regular; for all

i E I , all

&XI f = ~X1g

then

No analogous

simplification

known.

cylindric

Weak

Corollary

Proof.

Suppose X E A

f,g E ~U (p)

, there

Hence

[~N(F~X)]If=

I.I.14

,

X

' and all

g E CYYi , if

of the notion

set algebras

of regularity

are always

for arbitrary

Gws

regular:

Ws reg = Ws

that

~

is a

Ws

, f 6 X , g E ~U (p) is a finite

F ~ ~

[~(F~AX)]Ig,

w i t h unit element , and with so that

~X1f

= AX1g

(~NF)If

.

~U (p) Since

= (O~NF)Ig

g E c(FN~x)X

9

= X .

Thus by

is regular.

Corollary = Gs reg

f E X f3 ~Yi

g E X .

1.1.16.

A l s o assume

Gs

set algebras

1.1.17.

, and

Gws

If

~ < w

, then

Cs

= Cs reg

, Ws

= Ws reg

,

= Gws reg

reg Corollary

1.1.18.

If

~ ~ w

,

then

Ws ~reg c Gws

~

~ Crs reg

and

's is

~2

Relativization

1.1.19

Cs~ eg c Gs~ eg c Gws~ eg .

Proof.

To produce a member of

Gws reg N Ws reg , let

two disjoint sets with at least two elements, and let be arbitrary.

Let

~

be the full

Gws

U

p E ~U , q E

with unit element

Then it is easily checked, along the lines of the proof of E Gws reg ~ Ws

It is also clear that

the two-element Gws

with unit element

Crs

(provided that

Finally,

if

of the full

U

and

p

V

are as above and

Corollary 1.1.19. and

~V

If we let

~

that be

~ E Crs reg

For

~ m w

~

is the minimal subalgebra

~U U ~V

we have

, then

~ E Gs reg N Cs

Cs reg = Cs

, Gs reg c Gs

Gws reg c Gws

Proof. regular. Then

be

~U (p) U ~V (q) 1.1.16,

, we see that

V

has more than one element in its range).

with unit element

Gs

~ ~ Gs {p}

and

By

Let

~i

~2(P) E A

in a regular

Cs

I.I.Ii

it suffices to exhibit a

be the full and

Cs

A~2 (p) = 0 .

the only elements

with base

2 .

Hence

~2 (p)

X

such that

Cs

which is not

Let

p = <0 : K < ~ }

is not regular,, since AX = 0

are the

zero element and the unit element.

2.

Relativization

Our basic kinds of set algebras have been defined in terms of relativlzation of full cylindric set algebras.

We want to present here some results,

1.2.5, 1.2.9, and 1.2.11, which involve relativization and exhibit important connections between

Cs's

and

Gws

mentary characterization of unit sets of in 2.2.11.

's. Gws

First we establish an ele's, which was mentioned

I .2. i

Relativization T h e o r e m 1.2.1.

Let

~

F = [XEA

Under these premises,

13

be a

Cs

: sK}XAsKXX = X

cient for any given set

F=A

V

belongs

V

to

A

to

, assuming

imposed on

is necessary and suffi-

F : ~ ~ I

(so that in this case

); (ii)

V

(iii) ing

belongs V

to

belongs

to

A

and is a Cartesian A

space, assuming

and is the unit element of a

~ = 2 ;

Gws

, assum-

~ e 3 .

Under the same premises we obtain an additional (iv)

assuming

2 K ~ < m

are jointly equivalent a

o

assumption

(i)-(iii)

to belong

U , and let

for all K , ~ < ~ }

dependent on an additional

, each of the following conditions

(i)

with base

Cs

in case

Proof.

, the conditions

to the condition

~ = 2 , or a

Gs

of

(ii) .

~ • 3

and consider a set

a

, say

V = ~U'EI ~W~ pi)

i,J

I

sK~V .

with

i ~ J .

Now suppose

~.wj(Pj)

Pick a

is a finite

therefore we define for every

quently

i = j

conclude

that

where

and

such that g

of

V E F , as desired.

(i)

U .

is obvious, and so

n --j~w(PJ) = 0

of

(iii)

,

fKfk E ~W!pi)l

that

f = (~F)I

f

and

pi = ( ~ F ) I

g~ = f~

V ~ sSV N

E W i n W.. pj

for all

g E ~W~ pi) n ~W j(pj) Hence

for any

we clearly have

say,

by stipulating that

f E ~w~Pi) ~ V 9

VEF

w h i c h is the unit element of

and notice

~ E ~ ~ F ' we get

and

is the unit element of

the sufficiency

K,~ < ~ ,

(~F)~

C(o~,I)V = V

~ m 3 , with base

o~pi)

f E s~V N sKXV", thus ,

F ~ ~

g~ = f~

V E A

Given any

~ E ~ ~ [K,~]

V

in case

To establish

assume Gws

that

The necessity and sufficiency

is the sufficiency

conclusion:

f~K E There

.

If

~ E I~ 9

Conse-

V = sK~V N sK~V, and we

, and

14

Relativization

We wish now to demonstrate

1.2.1

the necessity

To this end, we show first that every set a

Gws

of both

V E F

(ii)

f E V

of

and N~meti.)

let

Yf = [ u E U

We need the following

If

f E V

For,

if

< < c~

(2)

If

f ~ V

(3)

If

f E V , u E Yf

For

we may assume

,

,

: fOEv}

facts about

(I)

Yf

then

f E ~ Yf

then

f E V = s0V

and

u E Yf

9

,

, then

, and

that

.

so

E V

and hence

Yf = Yf(0/u) _

K < ~

K ~ 0 .

fO f<

, then

Then

fKu E V .

f0 E V ~ s~V ~

f E V ~ s~V

whence

V

fK

, so

fOE E V .

~ ~V

=V

U

UU

O f~ E ~ V

~ce

fK E Yf

"

U

Also,

(iii).

is the unit element

(This portion of the proof is due to Andr~ka

For each

and

as

desired.

(4)

If

Indeed,

f E V , K < J , and

(4) clearly

the hypothesis have

fO,f0~V~ U

of (4). s~V

V

v E Yf(K/u)

follows Then

so that ~

u E Yf

from (2) if fKu E V

E =0

by (3).

fKu E V

.

From

Consider

(5)

~Y~f) t: V .

(6)

If

f,g Q V

and

f Q ~Y(g) g

(7)

If

f,g E V

and

~"Yf(f) ~ ~Y(g) g

yf c yf~K/u ~[~

~ ~0

now any

VH

we get

' then

(5)

Y g ~ Yf

, then

and

(6):

.

~'(f) ~f

= ~Y(g) g

B

, and assume

fOE E s ~ V n s O v = v

~rv

(4)

and

Assume

fOK,f OK E V . Hence UN

, as desired.

, then

v E Yf .

. We

Thus

1.2.2

Relativization

In fact,

(8)

f E

If

~,(f) if

f,g E V

Let indeed

By

(8) ,

In case

~ =2

f,g E V

then

yf ~ Y

~ 0

g

It follows (iv)

~f

and

Y

~ Yf

g

by

N

~y(g) g

.

Then by

is the unit element of a

, we have

~.~f(f) = ~yf

Gws

that

V

~Yf ~ ~y

~ 0 .

g

~ (f) = ~Y (g) ~f g

, as desired in (iii).

f EV 9

f0 0 n s0V 1 ~ V , and consequently, gl E SlV

and hence

Therefore

Furthermore,

C(a~I)V = 1

if

gl E Y f 9 Thus by (I),

~Yf = ~y

is a Cartesian space, as desired in

is clear since the condition

(g) g

Yf U Yg = Yh ' so that

by (7),

for all

= ~Y

~ (f) ~y(g) yf = g

(6) ,

~y(g) ~ ~ (h) Therefore, g Yh "

V

~y~f)

(6) , whence

~_ (f) ~y(g) yf A ~ 0 , then g

and

~. (f)

h E

~-(f) c ~y h) ~f -

, so that

18

g

by

(ii)

is equivalent

(8) Finally,

to

UpEvRgp = U . Theorem 1.2.2. with base a s~t

Assume

that

U , and let

~

~ ~ 3 . be a

Let

Crs

~

be the full

with unit set

Cs

V ~ ~U .

Then the following conditions are equivalent: (i)

~

(ii)

is a

Gws

s~V n s~V = V

Proof:

By

characterization For

~ ~ I

and

be the full ~(~)X

.

U1

~ ~ 3

(operating

Theorem 1.2.2

the corresponding for

of those elements of a U0

K,k ~ ~

of those unit elements of

On the other hand,

let

for all

in

~ ).

1.2.1.

Remark 1.2.3. For

Cs

;

Gws

's which are members of a

result is trivial,

in view of 1.1.8.

~ = 2 , there is no elementary characterization Cs 2

which are unit elements of a

be disjoint Cs 2

gives a simple elementary

In fact,

sets each having at least two elements.

with base

One easily checks

Gs 2

that

U o U U I , let 9

X = 2 U o U 2 U I , and let

is atomic and actually possesses

Let B= just

16

Relativization

three atoms

2

D01,X ~ DO1 , and

1.2.4

(U 0 U U I) ~ X

completely determined by the fact that

and that its structure

CoY = CIY = 1

It follows directly that there is an automorphism of changes

the atoms

interchanges

X ~ DO1

the elements

unit element of a all first-order

and X

and

(in arbitrary

of interest V ~ A

of such elements of

that if

~

sentences

Cs2's).

is a unit element of a

As

with field included in Im [AN3]

V

is the satisfies

DOI U ( 2 ( U o U U I) ~ X)

(or even higher-order)

with base

iff

Gs 2

X

that hold for all

Cs2's is possible.

Cs 2

is a full

As

DO1 U 2(U 0 U U I) ~ X

is not itself such a unit element, no elementary characterization

which inter-

DO1 U 2(U 0 U U I) - X .

(and indeed higher-order) Gs2's

~

Y .

2(U O U U 2) ~ X , and which therefore

Gs 2 , this implies that

unit elements of

for each atom

is

It is also

U , then an element

is an equivalence

relation

U . a m

it is shown t.hat for

terization of the unit elements of

Gs's

there is no elementary characwhich are members of a given

The other main results of this section are related to the following obvious

theorem.

Theorem 1.2.4.

similar to

For any algebra

CA

's the following

three conditions are equivalent: (i)

~ ~ Crs

(ii)

;

~ ~ ~L.~

for some full

Cs ~

and some

V s B ;

~ V

(iii)

~ C ~5..~ v

for some

Cs ~

and some

V E B.

The question naturally arises whether the condition can be replaced by general for

~ = ~L V~ .

is negative.

our classes

Ws

For arbitrary

~ ~ ~L@

C r s 's

in

(iii)

the answer in

Here we want to carefully consider this question '

Gs ~

, and

Gws

We begin with the following

Cs .

1.2.5

Relativization

simple

result.

T h e o r e m 1.2.5. Ws

with base

U

~(~)A

for

~

Proof. first note H~nce

Ws

if we let

V

~

that

Obviously

~ ( ~ ) V = 0 , and hence : XNV~A]

Remarks 1.2.6.

so

~i~

In fact, Thus

and let

~[ be the full

so that

V ( C .

9/

Gws

Cs

~ ~

~U~ q)

if

of

~D

i ~ j .

~V~D

in the subbase

~

.

with

U , we can take

For the converse

: XNV

s

E Su~ .

does not extend to

Let

V

~U p) U

tu ," note that

be the minimal 0

subalgebra

,

~i

U0 U U1 = w , of ~ [ V ~

(cf. 2.4.61),

~LV~

.

Thus

and hence

generated by its diagonal ele-

is any

but not in

Just constructed

sense.

elen~ent of

Cs ~

a

Cs

cl

with

V 6 D

and with

~D

is in

The Gws

following

for some

is

U 0 = 00 , U I = • , I , p = <0 : K < ~ >

c0(-v) N v = Uu[ q)

is included

~

, as desired.

1.2.5

It has characteristic

w , then in

9/ ~ ~ 6 V ~

with base

9]

On the. other hand,

so that

assuming

with base

~U p) N ~U q) = 0 .

Cs~

We let

let

is simply the B o o l e a n subalgebra

base

9/ = ~L 2

A ~ [XEC

The last part of

~ > o~ .

q = (I : ~ < ~ )

ments.

we have

be the full

V = ~U (p)

B C [XEC

is a

More specifically,

.

Say

Gws~'s with

R6Cs

and unit set

V E B ; in fact,

and

17

Let

widely-distributed

be a

(Pi) UiEl W i

is

We call

~

UO .

~ if

Usually a

Gws

J

Hence

9/ ~ ~L ~D . V

in that the subbase Gws

is normal,

, ~ ~ o~ , and suppose ~ (Pi) (P.) Wi n ~W i j = 0

W i = W. J

W i ~] W. = 0

A .

is unusual,

~ , with

normal if

,

or

whenever

W i [~ W. = 0 J

U1

in the

the unit

whenever

i,j E l

for all

i ~ J ; compressed ,

i,j if

;

and

18

Relativization

W i = W. J Cs

for all

i,j

Thus every

blished

in section

Gws

7

is a normal

~

is a direct

is a

P~Cs

It follows

of this p a p e r

widely-distributed

to a

Gs

Gws

factor of a

with

that

show below

in 1.2.9

It

is also clear

Cs

w h i c h has

Gws

at least

Gs

Gws

Gws

c P~Cs

Henkin,

lemmas

~ ~ w

, since

~

; so

by 2.2.10,

can always be

the same base as 3 ~ ~ < w

and

~ ~

~ ~ w

.

We

is a we have

the last part of 1.2.5 holds

then

Gs

~ ~ ~

that in case

of course,

, since we have

that

that every compressed

with

in case

Gws

~ Gws

= Gs

.)

We also have

1.2.9

is due

to

jointly by H e n k i n and Monk.

two lemmas about a r b i t r a r y recall

is i s o m o r p h i c

the same base as

having

1.2.11 has b e e n obtained

We n e e d these

(Thus,

~ < ~

in case

while

Cs

~

, we show in 1.2.11

= P~Cs

's with

from a

that this is true

On the other h a ~

Gws

same base.

o b t a i n e d by r e l a t i v i z a t i o n

for

, and every

from results w h i c h will be esta-

that e v e ~

We do not k n o w w h e t h e r a normal

Gs

Gws

~

is a c o m p r e s s e d

Gws

.

1.2.7

CA

~[b ~

's.

In c o n n e c t i o n w i t h

is a

CA

for e v e r y

kJ

~ CA R all

and every

K ,k <

~

Lemma

1.2.7.

~Lb ~

K,h < ~

(ii)

Let

for all

of the a l g e b r a

(i)

If

If

be any

K,k < c~ . .

, any

The

F C ~

F • A C ~

Let

sKbk " s~b = b

for

and

, let

b E A

, and assume

that

+i '_i 'c K' I

F,~ ~ ~

then b

b

etc.

conditions

9 C(FU[~})x

9 C(F)X =

O (F) I X

be the operations

(i)-(vi)

, and any

F ~ g = O , then

, then =

CA

following

finite

F U [F.] ~ cY

(iv) c(r)C l(r ) X If

SJ

F U & c ~

If

(iii)

(v)

that

.

sKb 9 sXb K = b

for any

h a v i n g the p r o p e r t y

b ~ A

x,y < b C(F)b

hold

then

:

9 C(A)b = b

.

~ cK(b 9 c(F)x ) 9

c(r)x , then

C(F)X

I 9 c(~)y = C ( F ~ a ) ( c (IF ) x 9 c(~)y)

1.2.7

Relativization (vi)

If

F c ~

Furthermore, K,~ < ~

let

,

eKk

!

then

assume

19

-C(F)X = C(F)(b .-e(F)X ) + -C(F)b

that

~ < w

c(~K,k])b.

and that

c(r

= 1 .

. For all

Then the following conditions (vii)-(xii) hold.

(vii) c(~.f~])b = I (viii) (ix) (x)

c(F)b = N~,~Ec~F e ~ e e

~k

= e

"

whenever

KX

~ < c~ and

~ @ K,~

.

dKk g eKk

(~i) d k = c(~{~,k})(b.dk) (xii)

Suppose

F U [~0 } c ~

that

~ E ~o- I , ~ E ~

w = c(r)x

, and

c o(C(F)X -~ ~,e.+-+t

Proof. for

F = 0

or

then

IF1 = 1 = IA I ; say

so

(i)

holds K E A

holds

for sets and

h E ~ N

ccru{~o} )x

by induction

A = 0 , so suppose F = [K]

in this case. F'

) =

(i)

b ~ c b 9 c b < c sKb K k K ).

, AI

with

(F U A )

b ~ c(r)b = C(F)b

= c(r)b = e(r)b

9

9

Then

E~..le~

- e~ 0~

(i). We prove

vious

9~

-%0 ~

"

on

and

A = [k}

.

IF U A 1 .

F ~ 0 # A 9 9

If

It is obIF@A1

= 2 ,

Then

9 cks~b = sKkb 9 skb = b K ' Now suppose IF'UAII

<

that

IFUA I > 2

IFUA[

Say

c(a)b ~ c(r)b

9

c(F)s~b b

s~(c(F)b c(A~K])b) 9

9 s~b g c(F)b

9 cKb = b .

9

and that

IA I > 1 .

Then

9 sKxC(F)b 9 sKc 9

, ~ 0 ~ F U [~I ..... ~ - I ] '

c(a)s~h

(i) Choose

20

Thus

Relativization

(i)

1.2.7

has been established.

(ii).

Assume

F U [K} c ~ .

We may assume

that

K ~ F .

Then

b 9 c(FU[K])X ~ cKb 9 cKc(F)x = cK(CKb 9 c(r)x) = cK(eKb 9 c(r)b 9 c(r)x)

= cm(b 9 c(r)x) and

(ii)

holds,

(iii). (iv).

(by (i))

This follows Obvious,

(v), Assume

from

(ii)

by an easy induction

on

Irl

by direct calculation.

that

F U & ~ ~ 9

Then

(by (iv))

c(r)x 9 c(a)y = c(r)x 9 c(A)c~a)y !

= c(rr~)(c(r)x c(A~F)c(&)Y) 9

9 c(A,~l~)C(A)y

= c (rr~)(e~r)~

9 c (r~' . )ClA)y )

=

(vi)

Assume

t

= C(r~)(C(r)X

F c ~ .

C(rFh)

(el

(~

X

(by (i))

.b)

(by (iii))

9 C s

(~)Y) 9

Then

c(r)(b 9 -c~r)x)

= c(F)(b 9 -(b 9 c(F)x)) = c(F)(b --c(r)x) = c(r)b

and

(vi)

follows,

Now we assume (vii)

since

c(F)x ~ c(F)b

the additional

We may assume

9 -c(r)x

that

.

premises

K @ 0 .

,

for

Then

(vii)-(ix).

(by (iii))

1.2.7

Relativization

0

21

0

I = s c(~l)b

= sKc(~{O,K})c 0

0

b

K

= c(~.{0,K])s
9 Sob)

= c (C~[K})(doK

" sKb " CKCo(doK " b))

= c (o~{K])(dO~

" b 9 c Co(do~

9 b))

c

(viii). ate consequence

(ix). (x).

ci .b ~ b (4)

Since

(v)

of

and

for all finite

(vii)

A = ~ , this is an In~nedi-

.

Obvious 9 We have, assuming

K ~ h ,

d KX = d K)~ ~ c (Ot"[K}) b =

dK~

(by (vii))

" c(~,~,~})c

c(o~K,~})(dKz

b

9 sK~b)

= c(~{~,X))(d~. ~ 9 b) ~ e

(xi).

It follows

(xll).

Assume

~cE~-,l

eXOXc

immediately

the hypothesis

from

(x)

of

(xll)

it suffices

enough

Also

let

z = c(F)x

"

Then

cX0z + cx0 w = C k o ( Z + W ) Hence

.

to show that

to show that

z 9 c

Co z " c

w = 0

= c X 0 c(F)x

.

w = 0 , and for this it is X0

or, by the definition

of

w

, to take

x0 any

~ E ~ ~

1

and show that

z 9 c 0(C(F)X .ex0%~ ) = 0 9

And in fact

22

Relativization

1.2.8

z 9 Cko(C(F)X 9 ekok

) ~ C(F)b

" -ekok

CkO(e(r)b9e(~qXO,X~])b) (by (v))

= c(r)b -exox~" c((ru{[Xo})"{x])b 9

= c(r_FkO,k

This completes

the proof of

Lemma 1.2.8. satisfy

b EA

(i)

(iii)

without

the following

c(o~l)h

"

-eko kg = O

.

9~

, and let

be a simple

CA

c~

Let

conditions:

for all

K,X < ~

9

for all

~,~ < ~

= I .

Setting

for every repeating

x K b , every

,

E r ~ 1 , and every sequence

k E ~

terms,

Cko(C(c~[ko]}X-~ where

1.2.7

3 ~ ~ < ~

s~b " s kb = b k

(li)

we have,

Let

])b

the sequences

<9_l-e z

and

) = z w

+ ~

(zK "-zK+ 1 "wK)

are uniquely

determined

,

by the following

stipulations:

zK

= c R (9 c )x (gX) ~<< (o~[X ] " ~<e<~

S

= [A : A ~ v - I ,

Igl =K]

for each

e

k~k9

)

for each

K ~ ~ ,

K < ~ ,

K wE = - ~

(~sEgc(c~[k

])c(~) , x "~g,gEA,~O

_ ekke)

(with

c(~) i

a~S K denoting

the generalized

cylindrification

in

N6b~

) for each

K < ~ .

1.2.8

Relativization

Under this hypothesis we have

Proof. of

1.2.7

By the assumptions

available.

d~%, c~, etc. and let of

A*

.

~ =3~gb~)B

(i)

and

.

Obviously A**

- , it suffice , then F = ~

A** ~ ) B

.

B~A**.

.

we have all the ~ r t s of

F~]

~gb ~

U {-eK

In view of 1.2.7

For any

by

-~,

:K,%<~ ] ,

+ .

to show that

-y E A**

y = I

1.2.7

(vi)

or

y = 0

we have

To show that

.

by simplicity,

so

(xi).

is closed under

y E A*

Suppose

B m A**

dKh E A * = A * * b y l . 2 . 7

A**

for every

E A*

(iii) it is now c.learly

and in fact to show that

K,k<~

is closed u n d e r

-y = eKk = c(~.~Kx})b

, then

then by

~-C~b~

be the set of all finite sums of finite products of elements

Obviously

Clearly

(li)

[c(r)x:xEBand

A* =

e~ongh to see how to prove the converse, E Su~

whenever

Again, we denote operations

Now let

A**

23

that

.

If

y =

y = c(r)x

-y E A**

.

If

.

If

F # ff ,

E A**

by

1.2.7

we have

-y = -C(F)X = C(F)(b .-C~F)X) + -c(F)b

and o b v i o u s l y (viii)

C(F)(b "

) E A* c A

while

-e(F)b

. Finally, we must show that

A**

is closed under each

we give an algebraic version

of an e l i m i n a t i o n

From the definition

we see that it suffices

A**

for every

with each for

x

y

of

A**

w h i c h is a product

E B , where

$,~ < w .

c

(Here

k

of quantifiers.)

of elements

to show that

of

A*

cky E

. Say

Clearly we may assume

that

L

D

each

we may assume

< ~ , and that

k E[K ~ 1 .

,~ ] Since

for each c(~k])b

I < ~

9

= 1 , by

Thus by 1.2.7

1.2.7

(vii)

(v)

, we

24

Relativization

m~y assume that

~ = 1 .

1.2.9

Thus we are concerned w i t h

y = c(&)x

where

x E B , ~ ~ A , and

< B < y .

If

k ~ ~

for each

A U [)~] ~ ~ , then by

9 ~ < e

w = c(A)x

of the form

9 ~ <~ - e

~ < ~ , and

1.2.7

(xii)

cry = C(~u[~})x 9 - c w where

y

~

~

in

case

we have

,

and D

e~w = ck(c(&)x" ~ < e = cXD = ck~

)

<~(c(&)x " C ( ~ k , ~ ~yC(~[~5])c

!

])b)

(&)x (by

1.2.7(v))

J

= r~ < c((~u{x]>.{~ ])= (A)X E A**

so that

cRY E A**

by premise z

(iii)

.

On the other hand,

of our le~na,

if

,

g U [X} = ~ , then

c y E A**

since under the n o t a t i o n used there each

is closed and hence is either

0

or

I

by the simplicity of

~

while

K obviously each

T h e o r e m 1.2.9.

In case

~

V

assuming ~ = ~6.~ v

be the full

Proof. and

A**

the proof.

, we have

let

is in

This completes

specifically, V

wK

~

to be a

~

also 2.3.14 or 2.3.15.

Cs

with base

It suffices

in place of

3 ~ ~ < w

for some Cs

.

Gs ~

we have with base with

b ).

U

~ P~Cs

U , we can take

and unit e ~ m e n t

of

~(~)A

for

1.2.8

It is easy to see that

Condition 1.2.8

More

V E B ; in fact, if we

to v e r i f y the hypotheses

and

Gs

~

~

.

(with is simple; see

(i) follows from 1.2.2, w h i l e 1.2.8

1.2.9

Relativization

(if)

is obvious.

place of

b ).

Say

.

For each

U = ~iEiWi in(u)

(I)

Assume

, such that

for every

the notation of

V = ~iEi~Wi u E U

u E Wi .

K,X < ~

25

where

1.2.8

(ill)

W i N W] = 0

there is a unique

(with for

i E I

V

in

i ~ j .

Thus

denoted by

Thus clearly

and

f E ~U

, we have

f E e

iff

Kk

in(fK) =

in(f k ) 9

Choosing any

(2)

x ~ V , we then get

for every

K < ~

and

f E ~U , f E c ( ~ K ] ) x

(Recall from the Preliminaries every (3)

f E ~U

pJ~x = (g% :g Ex]

f .)

E pjSx Hence for

,

f E ck0(c(~X0])x

such that

that

iff

in(u) 9 i n ( f

9 [ Ev~I - ex0k )

for all

)

iff there is a

u ~ pj~0 x

~ E ~ ~ 1 .

On the other hand, obviously

(4)

elements

K ~ ~ , z = ~U or z = 0 ; z = ~ U iff there are K < E * . w 0 ~ pJkoX .... ,wK_ I E pjkK_]x such that in(w ) ~ in(w )

whenever

~ ~ 0 .

that

(5)

for each

Since

in p j . x = in*pj"x P

for all

x c V

to rewrite

(4)

for each

< ~ v ,

z K = ~U

as

or

zK = 0 ;

I[ in(u) : u ~ pj~0 x] 1 ~ ~ (6)

c(~)

,

~

= ~J

{~w i

: iEl

, %f

K,p < ~ , we may use the fact

nx#O]

9

zK = ~U

iff

26

Relativization

One can easily prove

(6)

by observing

(~) X = Co(C(~f~l~X7 "V) 9 V

and using

C ~

(7)

if

N < ~ ,

1.2.10

A ~ ~ ~ 1 ,

from (2)

IAI = K

and

1.2.7

(iii)

From

(6)

f E ~U

that we easily get

, then

!

f E ~ Egc(~.~[~ ])c(~)x distinct

Hence

(8)

~ , 8 ~ A ~ and

for any

f E wE

f~ ~

f E ~U

iff

" ~,SE&,~#~

- e~Lk8

iff

[ Wi : i E l , ~ W i

i n ( f ~) # i n ( f

~x#O}

for all

) for all

~ EA

.

we have

l{in(f 1 ) ..... in(f)~

)} N [i : ~Wi A x # O ] I

< K 9

~-I Now let us compare C~o(C(~ko})X Suppose

z

(3),

(5), and

(8)

.

" ~ Ev~ I - e Ok ) = ~U

=0,

If

z

= ~U

, we see that

also,

and

1.2.8

pj

x}l

.By(5)

and let K = l [ i n ( u ) : u 6

(iii)

holds.

we have K < ~

9

X0 Clearly

{in(u) : u E p j ~ o X } = [i : ~ W i ~ I x # O ]

z

+

~ ~<~ z~ -- Z~+l

Remark

Ev..1- e Ok ) .

the proof.

1.2.10.

case

J = 2 .

Lemma

1.2.8

but with

that

"w ~ =w K

= c o(e(c~[XO })x.r; ~lis completes

It follows

The preceding

To cover having

premises

theorem can be extended

that case we would

the same conclusion (i)

(I)

0b o 1 b ~ sI sob

(2)

dol ~ b

(3)

c g (c x "b)

and

(ii)

first establish

and same

replaced

to the an analog

premise

(iii)

of ,

by the following:

;

9 b < c (CKX. b)

for any

k,K < 2

and

x ~ b .

1.2.11

Relativization

27

The proof of such a lemma for the case version of the proof given for

1.2.8 .

there is no

difficulty in extending

by verifying

that the above hypotheses

the conditions assumed in For

~ = 2 ,

1.2.9

~ = 2

With such a lemma at hand,

1.2.9

[HR]

to cover the case

(I), (2),

and

(3)

~ = 2 ,

hold under

1.2.9 . specializes

to

paper of this series we shall show that results from

is merely a simplified

Gs 2 c P~Cs 2 .

In a later

CA 2 < IP~Cs 2 .

In view of

this sugges~ that perhaps

Crs 2 = P~Cs 2 , but

this has been disproved by Andrlka and N~meti~ see 2.3 of [AN3] (p.155).

Theorem 1.2.11.

In case

~ m w

we have

Gws

~ RLCs

(pi) Proof.

Let

be a

Gws~

(pi) (pj) Wi ~ ~Wj = 0

where

be the base of Let

~

~

~

claim that

for distinct

and choose

be the full ~ = ~V ~

Cs~ .

with unit set

UI

with base

V =

i,j E I .

such that

UiE I

Let

Ut ~ U

U I , and let

W.l

'

U =UiEiWi

is infinite.

~ =|

.

We

To prove this, we shall again follow an

elimination of quantifiers argument, as in the proof of

1.2.8

.

Let

C* = [C(F)x : x E A, F finite subset of

Let

C**

members of

a finite subset of

~} U [-C(r)V : F

~] U {DKx : K,~ < ~} U [-D<~ : ~,~ < ~} .

be the set of all finite unions of finite intersections C* .

a

Obviously

C** ~ Sg(~)A

,

of

and it suffices to prove

28

Relativization

the converse, for all C**

i.e.,

that

A & C** ~ Suq .

K,X < ~ , and

C**

is closed under

y ~ C*

-y ~ C**

for

for

9

by

y = C(F)x

We may assume

A = C**

that

y

1.2.7

CKy s C**

-y s C**

-C(F)V,DKk

(vi)

, DKX s C**

To show that

to show that

of the form

y

So it remains to show that y ~ C**

Obviously

is closed under unions.

, it suffices

This is obvious

while

1.2.11

for each

or

-DKx

,

, w h i c h applies by 1.2.1

whenever

K < ~

and

is merely a product of members of

C* , say

y = C(ro)X 0 N "'" N C ( r ~ _ l ) ~ _ I

N-C(Ao)V N n

""

N...

N

-C(A _I)V

RD

DX0~ 0

k6_ i~6_ 1

n -D~o00 n "'" ~ - D ~ _ i P ~ _ l

where < ~

~,7,6,~ < w , ,

@ < ~

we may assume K E [ ~ ,~&] Say

,

and

that for all

F

and

A@

are finite subsets of

k ,~ ,~6,p@ < ~ g

K ~ F

L

5 < 6

for all , and

for

I < 8 ,

~

for

~ < 6 ,

e < s .

Further,

K ~ A

for all

~ < ~

~ 6 [~@,P@}

for all

,

@ <

K = ~0 = ''" = ~6-i = P0 = "'' = P~-I

Further, we may assume that the other and from verified

K , and similarly

~

's

are distinct

for the

facts show that we can assume

that

6 's . 6 = 0

from each The following easily (here

K ~ ~ ):

c (x NY N D< ) = c (x N D ) Rc (Y N D~ ) ; C ( C ( F ) x 5 N DK ) = C(FsU[K])(xl

c (c(r)x c n D <

R DK )

if

"~ 6 F

) = C(r~{~])U{~})(D~ n C x

nv)

if

'~ E F L ;

.

1.2.11

Relativization

CK(-C(A

)V N D )

29

= -C(( A .[~])U[K])(DK~

Q V) 6 C**

D

, proved above;

by closure under CK(DK~ N C(-D

Also, by

1.2.7 (v)

our considerations

DK~) = D

,

= -D

N D

we can assume that to

y

8 g 1 .

Thus we have reduced

of the form

C(r)x n -c(A0)v N .-. n -c(av_l)v N -DKo 0 N

where 'the

K ~ F , p's

appear,

for each

7 = 0

KPs I K # Pl

~ < ~ ,

or

C(v)x is a factor of

s = 0 .

proof of

N -D

for each

are distinct, and where perhaps the factor and possibly

Case I. take

K ~ A5

"'"

s = 0 .

y .

does not

Now we consider two cases.

We proceed by induction on

Clearly we may assume

1.2.7 (xii)

C(F)x

~ < s , all

that

7 > 0 .

~ .

First

Then, as in the

we see that

cKy = C(ru[K])x 9 -c (c(r)x n ~
1.2.7 (v)

> 0 .

For each

(9)

and

(vi)

~ < c

CKY E C** .

we have

Now assume,

inductively,

C(F)x N -C(FN[pl])V = -DKpl

c~y =c (C(r)~ nn<~

that Thus

c(~)v nr~<~ - c(r~{%])v)

U ct(c(F)x N f~
- D <0 L

n [i<Ec(r~<~})v)

.

No~ we have CK(C(r)x nc/<~- c(a)v n f/<~ - c(c~(%})v) 6 c** case

~ = 0

Just disposed of.

And for each

~ < s

we have

by the

30

Relativization

c~(c(r)x n ~

1.2.12

- c(~)v n ~<~ - D ~

= CK(C(F~{p

])(C~r)x ) n ~ < ~

n c(r..4; ])v)

- C(A )V n ~ < E

- DKO )

by 1.2.7

~<~,~#

-D p ) ,

to which the induction hypothesis applies.

Cas___ee2.

C(F)X

(*)

is not a factor of

c (~
In fact,

let

clearly

uK E Q<~

By proof of

u ~ ~U'

s

(*)

-

.

Choose

we clearly have

1.2.11

Remark such that Ows

Thus

- D

For every

For each

is not a Gs

Cs

above

facts are trivial.

CA

) ==u' (U lj [up ) , and

l

: g < ~])

(*)

.

Then

follows.

in this case also,

,

, since

is a

CA

we have

there is a

so the

Gs

~ Gws

Cs ~

~ P~Cs

and a

to

IGs

It

~ IP~Cs

such

~

D01 ~ V = 0

and

, and V .

IGs

For example,

V = [f E ~ 2 : f 0 * fl] (see

It is of more interest (~ince

IGws C I R ~ C s

2.2.3

(ii)).

2.6.42

See

let Then

So the

to construct an example

is the class of all representable

~s, this will provide another example of a non-representable

we gave one in

V E A

, and in fact is not even isomorphic

with base 2, and let

is not a

~.~ v

CA~

~ ~ 3

~ e 2

Gws

C P~Cs

9 ~V~

in which

Kp~

CKY E C**

is a trivial matter to construct be the full

Kp~

is complete.

1.2.13.

~,.~ v

Then

s E U' ~

C ( & )V N ~ < ~

Corollar X 1.2.12.

a

y .

(v)

CA

;

1.1.13 for the notion of a representable

1.2.13

Relativization

CA

; it w i l l be proved

class

of all such.)

Theorem such that

~ V ~ E CA

and let

~

of our

Cs ~

f

of

while

U

Let

For each

cf

I~ I

is to be

~ U W

Let

for e a c h

for some

equivalence

G

relation

that

A

be the cylindric gu ~ u

if

on

.

condition , and

2.2.3

Let

on

, and so there

~

is due to Henkin.

~

and a

V E A

with

onto

to show Since

is clear. ~ V

D

Clearly

G

is the universe

g,h E ~U

classes

implies

that

is a

b E U

with

U

u ~ K~

,

of all

sets

It is e a s i l y

~U

We let

K < ~

u E U

let

We now set

U (<0',I,2 .... >/---) .

that

~ ~V ~

c V

there

iff

is an

---

of

, of

g -= h

=-

For each

is a

CA

, 9 and to do

for all distinct that

We may assume

h

that

under

A

~ f ~ T

define

be the c o l l e c t i o n

Kk

,

The base

we have

N o w suppose .

W

~

w i t h universe if

~ N W ~ 0

of

is n o t one-to-one}

h E CK(CkX NV)

our a s s u m p t i o n

~

field of subsets

gu = K

2.2.3.

(ii)

A

of equivalence

Now we want

this we shall apply

is the

be the set of all p e r m u t a t i o n s

N o w for

U

, and

:g ~ f

Cs

It is e a s i l y v e r i f i e d

set algebra

u E ~

V E A

U

is a cylindric

V = [fE~U

Clearly

K < ~

f E G

w h i c h are unions

verified

N V

mapping

such that for some p e r m u t a t i o n

f og ~ h

IGs

0~

function

of

that

shall now give

be a set of cardinality

be a one-to-one

~ ~ 2

easily)

is a

a group of p e r m u t a t i o n s

X ~ V

(rather

there

,~ IGws

W

fK t ~ (~K) t

X = ~U

1.6.3

The e x a m p l e w h i c h we

1.2.14.

Proof.

in

31

is an

h K~ab E X .

K,h < ~ that

, X E A

K # X 9

a E U If

K,~ < ~

with

h bX E V

,

Now hK E C X a k

, then

,

32

Relativization

hbk 6 C K X

n V

h Ib E V

and hence

g o h~

(2)

hk # b

For,

otherwise

(3)

hK ~ a

For,

otherwise

Case

I.

, as d e s i r e d .

Hence

suppose

is o n e - t o - o n e . .

h~ = h E V

.

.

h bk = h ~

ga = gb

.

, and

let

b ( W

Then

h 6 Ck(CKXnV)

Thus

(I)

or

1.2.14

h cd kK E X

6 V

Let

N o w we c o n s i d e r

c = ghK

d = ghK

since

or

h KX 6 X ab

gc = gh<

, so

Case

2.

gb # ga # ghK

h E Cx(CKXAV)

and

gb # ga

Let

c = ghK

.

And if

.

(ghK)'

(ghK) ~

and

cases.

according

according

h XK h KX cd -= a b

as

and

'

as

b 6 c~

a 6 (~

hX E V c

or

a 6 W

since

9

Then

also

or

several

by

(I)

gh< ~ gh~

,

gb # gh~

for a l l

b E 0~ , c = (ghK) I

if

for a l l

~ # K,X

b E W

.

b # K,k

and

Then

ghK # ga

h XK E X

.

since

ca

h ~K ba E X

and

hk E V

since

gc = ghK

so

c

Case by

3.

gb # ga = ghK

(I)

, we have , so

Case

gb # ga = ghK

Case

5.

h k K a'(gb)

Thus of a

Gws

.

, h

ak

~V~/

6 V is a

To s h o w

9

Hence

k hg a E V , k = 0 .

that

Thus

since

h XK 6 X ab

Again , so

by

g o h Kk ab

and

is o n e - t o - o n e

hk 6 V

'

since

a

a 6 ~

,

a E W

and

and

b E ~

-

.

b E W

, so

.

It is o b v i o u s R2~Vgl

(I)

h E Cx(CKX NV)

Then

, h E Ck(CKXNV) CA

~

.

, K = 0 .

and

gb ~ ga = ghK 6 X

r ~

h s Ck(CKX NV)

h ~ K (ga)b , E X

Hence

0 ( [K,k]

a,b

ga = ghK 4.

h E Ck(CKXnV)

~

is n o t e v e n

that

V

is n o t

isomorphic

to a

the u n i t Gws

set

, we

.

1.3.1

Change

shall use

the equation

(4)

to check

that

holds

(4) ~

itself,

(5)

NOW

2.6.42

cl(Y" c0(clY.-y))

It is easy

in

of

of base

that this equation

in

~gV ~

for any

is to say

X = V

such

Thus

(5)

is an

with

in

~

there

f2~ E X .

or

a = I'

and

the second

(5)

that

From

and

.

that

X 6 A

Let

a E U

But

possibility in

~

So

~V

equation

holds

f = (0',01,2,3,4 .... ) N V

,

with

and ~XNV))

f0 a 6 N DOI

it follows

is not

Then

1 0 1 f01'l = . ~ V

that

Hence

yields

isomorphic

if

' and a

a = 0'

f0 = (i,,01,2,3,4,...> a ~

To say

) = 0 .

the first p o s s i b i l i t y

yields .

X

Gws

:

f s CI(X N C 0 ( C I X N

the form of

b = I .

does not hold

in every

the following

fol'l 0 = E ~ X


holds

N V N -C0(CIXN~DoINV

X = <0',0,2,3,4 .... >/m

fol E X , and

:

9 -c0(clY 9 -d01 ) = 0

CI(XNC0(CIXN~XNV))

let

33

b 6 U

and

b = 0 ,

f0 a = f E D01 6 V

.

to a

Thus

Gws

3

G i v e n a set algebra 9/ isomorphic IUI = IWI

to a set algebra

, where

Theorem

the a n s w e r

1.3.1.

(i) f

element

V

.

Suppose

For any

X E A

let

phism

from

~

9/

onto a

w i t h base

U , and given

w i t h base

W?

is obviously

Let

9/

is a one-one

FX = [y s ~W : Crs

be a

~

We

first consider

w i t h base

function

w i t h base

set

W

, is

the case

yes.

Crs

f-1 o y

some

~ X} W

from .

Then

U

U

and unit

onto a set F

W

is an isomor-

and unit element

FV

.

.

'

34

Change of base

(ii)

If in

(i)

~

is a

Cs

(iii)

If in

(i)

~

is a

Ws

is a

Ws

If in

S. N S. = 0 l j

for

U i E i f f f * S i , where (v) where

~

is a

i ~ j , then

Gs ~

(i)

is a

UiEl~(f*si )(f~

~

is a

Cs

with unit element

~U (p)

with unit element

~iEl Si ' where v

is a

f S i N f Sj = 0

~S (pi) N ~S (pj) = 0 i i

element for

If in

(i)

, then

for

for

Gws

Gs

with unit element

i r j . with unit element

i ~ j , then

, where

~

is a

~(f*Si)(f~

II ~(pi)'" ~iEl ~i Gws

with unit

N ~(f*Sj) (f~

the function

F

true, e.g., if

1.3.1

to the special case

~

is a full

is rather extensive.

Remark

U = W , in some cases

is an automorphism of the algebra

general kind of Galois theory.

1.3.2.

Cs

~

.

This is always

In this way one can develop a We shall not go into this theory, which

See, e.g., Daigneault

[D ] .

The less trivial question concerning change of

base arises when the two bases have different cardinalities. our discussion of this case we shall show that for each

base.

= 0

i ~ j .

If we apply

is a

then

Uw(f~

with unit element

(iv)

1.3.2

Cs

with an infinite base not isomorphic to a

First suppose that

equation

~

3 ~ a < w 9

holds identically in every

fails in some finite

Cs

Cs

To begin

~ ~ 3

there

with a finite

Then, we claim, the following Cs

with a finite base, but

with an infinite base

(this equation is due

to Andr~ka and N~meti, and replaces a longer one originally found for this purpose):

1.3.2

35

Change of base

c(~)[-c(~l)X

+ c(a N 2 ) x . d 0 1

0 1

+ c(~2

~ 9 SlS2C(~ N 2

I 9 -s2c(~N2)x]

To show this,

let

be a

hold identically.

Cs

w i t h base

U

)x

= 1 .

in w h i c h

does not

Then it is easy to see that there is an

XEA

such that the following conditions hold:

(I)

C ( ~ ~ z)X = ~U

(2)

C(~N2)X

(3)

C(c~2)X

N o w let

;

, 0 1 N SlS2C(c~.2)X

c

--DoI

R = [21u : u E X}

I -c s2C(c~.2)X

.

Then

R

.

is a binary relation on

U

satis-

fying the following conditions:

(4)

for all

u E U

there is a

(5)

for all

u E U , not

(6)

R

v E U

(uRu)

with

uRv

;

;

is transitive.

It follows

that

U

is infinite.

Thus

E

holds in every

Cs

with a

finite base. Now we construct a finite fails to hold in tional numbers and discourse Then

<

language for

.

Let

~

B .

we see that

~

with an infinite base such that

~ =
, where

is the usual ordering on ~

, w i t h a sequence

[~C~)- : q0 a formula of A]

finite base in

~

Cs

B B .

is the universe

v 0 < v I , we see that

is finite. ~

fails in

Letting ~ .

Let

(v~ :~ < ~> of a

F r o m the usual decision procedure ~

is the set of ra-

X = ~(~)

Cs

A

be a

of variables. ~

with in-

for sentences holding , ~p

the formula

36

Change

For Theorem

~ > w

1.3.3,

9/

Finally, 0 .

If

9/

Cs I

we shall Gs 2

ua E U let

with

show

(resp.

Cs 2)

9/>',-~ .

If

for some

subbase

Let

~

.

Let

In fact,

some

P E ~W

onto

W

(I)

also holds

of

IWI

D~

~W (p)

, and any

in

1.3.4.

~

with

of

9/ .

Let f

to a

has base Cs I

with

For each n o n - z e r o

U' = [u a : a E A}

Then

.

In a later article

Gs 2 (resp.

is an i s o m o r p h i s m

Cs2)

, and

of

9/

in this series

is isomorphic

to a

base.

and

.

9

be

Gws's W

and assume of

~

that

, then

IWI = IW'I

Then

~ 0

is included q E ~W (p) of the

9/ , and

in the unit element

such

this gives

Namely,

that

KI q

left side of

The i m p l i c a t i o n

9/ > ~

Crs 0

.

C~ ~ < ~ D

in the other direction. Cs

U'

Any

is isomorphic

.

By

w i t h a finite base.

~ < 2 .

E a

base.

9/ .

w i l l be a m e m b e r

Remark

a

~

Cs

for some subbase

< =

N~<~< x

in

base

~



w i t h a finite

W'

to a

is the base

fa = a N Iut

lWl < ~ n w

Proof.

U

so that

1.3.3.

w i t h an infinite

on the case

say

finite

1.3.3

Cs

isomorphic

that any finite

Theorem

(1)

be any

Cs I , then

In fact,

a E A

onto a

is not

is a finite

choose

for a n y

9/

some remarks

a finite base. a E A

, let

of base

(I)

.

the desired

in T h e o r e m

for each

, and a subbase

maps

W

of

9/

K

9

set

W'

that

.

does not hold

there are a with

for

one-one

It follows

1.3.3

~ > 2

of

Gs

71 ,

IWI < ~ N 0~ ,

c~ such that objects,

the base of let

U

and

~ W

has c a r d i n a l i t y be disjoint

~ Iwl

sets w i t h

To construct IUI >

IWl = I .

these Let

1.3.5

Change of base

be the full

Gs

with unit element

with unit element fX = X n ~U

37

~U .

~ U U ~W , and

The mapping

is a homomorphism by

f

from

2.2.12.

A

~

onto

the full B

Cs

such that

Clearly the above properties

hold.

We can strengthen a part of

1.3.3

by use of the following

notion of base-isomorphism.

Definition 1.3.5. by

~

the function (ii)

elements

Let V

~

and

F

With

f

defined in

and Y

(1)

9

be

as in Theorem 1.3.1, we denote

1.3.1.

Crs's

respectively.

with bases

We say that

isomorphic if there is a one-one function is an isomorphism from

~

onto

9

f

U ~

and and

mapping

W

~ U

and unit are base-

onto

W

such

.

The following result is an algebraic version of ~he logical result according to which any two elementarily equivalent finite structures are isomorphic;

it is due to Monk.

Theorem 1.3.6. regular

Cs's

has power

Proof. bases

U

and

Let

~

and

, and assume that < ~ n w , then

~

Claim.

be locally finite-dlmensional

~ ~ ~ and

.

~

If the base of either

W

respectively, and that ~

onto

~

9

~

and

IUI = ~ < ~ N ~ . .

Note that by

9 Let

have G

1.3.3,

Now we need the following

For each regular

Cs

c~

~

or

are base-isomorphic,

Assume the hypotheses, and suppose that

be the given isomorphism from IWI = IUI .

9

having a base

T

of power

< ~ N

38

Change

there is a function s

of

T

C

into

(a) foT

S

C

of base

w h i c h assigns

1.3.6

to each

T E

~

a mapping

such that:

for any

x E C

and any

f s ~T

we have

f E ~s~x iff T

~ x ; (b)

if

F

is an isomorphism

w i t h base of power E~

< ~ n ~ , then

from

~

Fs~x=s T

onto any regular Fx

for all

x E C

and

.

This c l a i m is of course related to our considerations I.II.

For

In fact, let

~ ~ w

x s C

, and let

let

k0 =

s

.

k~_ I ...

f E s~y

k _1

f~ I

K 0 < . . . < Ks_ I , ordinals

be the first

K0

... s

TK~_ I sk 0

iff

, with

<

Then we set

s

TK 0

Now note that

~x = [~0 ..... K _I]

k0 <...<

Ax U [TK 0 .... ,TK _I]

s~x

in section

it follows e a s i l y by the methods of section I.II.

for any

7 E ~

not in

E y

K~_ I x . kl_ I

for all

~,9 <

and all

The c l a i m now follows easily, using the regularity of

(a)

Cs

y E C .

in checking

.

To establish

the c l a i m in the case

~ <

is a transposition

it suffices

sider the case in w h i c h

T

(see the preliminaries),

since every transformation

position of a finite sequence of replacements this ca~e we let

[K/k,k/~] of

~

to con-

with

K # k

is the com-

and transpositions.

In

1.3.6

39

Change of base

U

(sVs~skx n DKV)

U ~V[),9#K

= (s~sksKsVx N D ) ,~;~ k K V ~ V~

Then condition and

(b)

x ~ C

(I)

if

f E DKX

, then

(2)

if

f E Dk~

with

V # K,k

, then

f o 'r = f ~ [ V / k ]

o [k/K]

o [K/V]

;

(3)

if

f E D

with

V ~ K,k

, then

f o 9 =

o [~/x]

o [x/v]

;

(4)

if

f 6 D

with

~,~ ~ K,k

D

f = f oT ;

f E s~x~ .

= ~T

f o~ E x .

suppose , and so

f E D

f o [~/~]

V ~ ~ , then

and

(I) ; the other possibilities

by

Second,

U

First note:

(a), first suppose

N o w to v e r i f y

(4) o

KV

.

(a),

let

f oT 6 x

f s aT

For condition

of the c l a i m is clear.

Since

If

f o~ = f~

f E x N DKk

~

, then

are taken care of by (2)IT1 < ~ , we have

for some distinct

~,~ < ~ .

Then

W

(1)-(4)

f E s~x , as desired.

yield

We have now fully established

the claim. N o w let sh~

that

in w h i c h and

X

w e set

~ ~

u and

be a one-to-one ~

function mapping

are b a s e - i s o m o r p h i c

is finitely generated.

is finite.

For each

A

:

; thus

onto

~ .

To

we shall first take the case ~ = SgX , where

w mapping a subset

w ~ = w U

the following element of

Say

U

w' E ~ U

F

of .

0 ~ X = A ~

into

N o w consider

U

40

Change

1.3.6

NCsuo w,x :x ~ x, w E ~XU, w' E x]

(5) N

N{~Suow,X :x

N

N K < ~ < ~ N DKx N C~

(This element finite = w ~ h

of base

structure.) w

is a m e m b e r

Applying

G

UK<sDK~

is an a l g e b r a i c

for each

Let in

expression

h = u

-I

(5)

element

n[~ow,GX

U <

u-10

of a complete

diagram

:K E ~ >

Then

(5), so it follows

of the element

to

the following

(6)

~ x, w ~ ~Xu, w' ~ x]

(5), w h i c h

and using of

B

(b)

from

9

(a)

of a

h ~ u ~w

of the claim

is thus shown

l

that

to be non-zero.

of the claim we conclude

that

is non-zero:

: x E X, w E ~ U ,

w' E x]

n N{~~UeW ,G~:xEX, w ~ ,

w' Cx]

N N~<.
g E ~W

g ou

be any m e m b e r

is a one-to-one

is an i s o m o r p h i s m 0.2.14

(iii)

-I

So, Let

=k E x .

~

let

now

w = ~xl

g~

.

Now

we see that

Fx = Gx

mapping

to show

( ~ I o k)

is a member

Gx

It is easily U

~x

k E Gx

1g .

of

.

(6)

onto

Cs that

x E X , and suppose

g

E Gx

.

into the full

Thus because ~

(6)

function

it suffices

as desired). f

of

of

checked

W

.

x

k E ~x

Similarly,

1k

.

Thus by

(hence .

is regular,

we infer using

~ u = w' = ~ x

W

XI ~ = Xl G

Since

f =

By 1.3.1,

w i t h base

that

that

~ = G ,

Thus ws E x

the c l a i m

that

, so by the regularity

k ~ Fx

implies

k ~ Gx

, so

.

Having the general

taken care of the finitely case.

Let

generated

$ = {C : 0 ~ C = A,

case, we turn to

ICI < w}

.

For each

of

.

1.3.7

Change of base

C E ~ and

let ~

I C = If : f

4~

is a one-to-one

is a b a s e - i s o m o r p h i s m

of ~ f C

function mapping

onto a subalgebra of

the finitely generated case treated above shows that all

C E N 9

IICo I ICoWD of

Since each set

minimum. =

~

For each

we have

Therefore any member of

IC 0 onto

9

improvements

of

1.3.7-1.3.10,

and address

Let

is regular,

Lenmm 1.3.8.

Thus

CO E $

for with

, and hence

induces a b a s e - i s o m o r p h i s m

and Remark 1.3.11, are due

the question concerning possible

~

and

~

then so is

be b a s e - i s o m o r p h i c

Crs's

via

F .

Fx .

Let

~

be the full

be a regular element of

~

not in the minimal

9

~}

1.3.6.

L e m m a 1.3.7.

l~xl < w

W ,

IC ~ 0

ICoUD ~ ICo

IC 0

onto

, as desired.

to A n d r 4 k a and N4meti,

x E A

is finite, choose

D E ~

The next few results

If

IC

U

Cs

with base

Then there is a b a s e - a u t o m o r p h i s m

U .

subalgebra of F

of

~

Let ~

x

, with

such that

Fx # x .

Proof.

For each equivalence

m E = [~{DK~ : ~ E k}

For each

s s ~U , let

y

we have

t E y -~ x .

tK = tk

iff

on

&x , let

~ [~{--D~k : K,k E ~x, K ~ k}

=

Say

sK = ski

Then we put

UsE x rues"

is in the minimal subalgebra of

there is a

E

Es = [ : K,k ~ ~x,

y

Thus

relation

t 6 rues ,

sK = sk .

~

.

Clearly

s E x .

x _c y , so

Then for all

Hence there is a p e r m u t a t i o n

<,k E A x f

of

42

Change

U

such that

p h i s m of ~xl

t ,

x ~x

.

Since

for all s C x

t ~ x , and

~x

Lemma

1.3.9.

,

is regular,

algebra

x of

that

~

.

fx

K E &x

, we have

is regular

Let

Then

Let

~ ~ ~

Thus

~

f ~ s E ~x

is a b a s e - a u t o m o r -

.

(cf. 1.3.7),

U

such that

Fx ~ x .

.

easy to v e r i f y .

Now so

~xl

( f o s) =

f o s ~ x .

Let

that

G

since

~J~

(~U~U

~

Thus

that

is not in the minimal

let

F

f

of

let

~

~

onto a

(q))

.

s) U

to check Let

and

be the

subCs

full

be a b a s e - a u t o m o r p h i s m

be arbitrary. .

For any

Because

that

of Gx

Cs

01

of

c E C

let

~ ( ~ U (q)) = 0 , it is ~

.

Let

f = A I G

is n o t regular.

z E ~ U N ~U (q)

be a r b i t r a r y

and

Since (note

that

Now we set

(~~~x)Iz

,

q' = ( ~ x l s ) U ( ~ x ) l q z ~ E C~U ~ ~U ( q )

, and

is an e n d o m o r p h i s m

0).

(~xl

of

q E ~U

only

s E FxNx

z~ =

Thus

x

, and suppose

is a h o m o m o r p h i s m

By Len~na 1.3.8,

It remains

Fx ~ x , say IUI > I

~ E Cs

, and

be the base

Gc = (c n ~U (q)) U ( F c n

= f*~

, let

l~xl < ~

there

U

w i t h base

and

.

is not regular.

Proof.

Fx

1.3.9

, as desired.

x E A

such

~

fsK = tK

of base

.

q~ E ~U ( q )

are

regular

(cf.

1.3.7),

we h a v e

ql

~ Gx .

But

A x l z ~ = &x~ q~

.

Furthermore,

z~,q ~ ~ Fx and

gGx c

TM

x

Ax .

since .

x

Hence Thus

and z ~ E Gx

Gx

is

not

regular.

Theorem finite

1.3.10.

dimensional

Let

regular

~ ~ m Cs

. ~

T h e n every n o n - m i n i m a l is isomorphic

locally

to a non-regular

Cs

1.3.11

Change of base

Proof. direct).

By 1.5.2 (i) below,

that for each ~

~

morphic.

From Lemma 1.3.7 and Theorem 1.3.10 it follows

~ a ~

there exist locally finite-dimensional isomorphic

with finite bases,

~

regular, which are not base-iso-

Thus regularity cannot be dropped, for

Theorem 1.3.6.

(I)

is simple (the proof is easy and

Hence the theorem follows from Lemma 1.3.9.

Remark 1.3.11.

Cs's

~

43

~

or

~

, in

Andr4ka and N~meti have also proved the following:

In 1.3.6 we cannot replace

~

ELf

by

~,~ E Dc

by

~

; see

[AN3] , Prop. 3.5(iv). (2)

In 1.3.6 we cannot replace

(3)

The condition that one of the bases is finite cannot be removed.

They also noted that for

N,~ E C s

~ ~ w

Cs

Ws

(or

and

q = <0 : K even < ~> U
and let

~

we have

~ > ~ , and since

~ ~ .

cannot be replaced by

Gws ) ; we give the simple example.

minimal

with unit elements

be the minimal

Similarly,

Cs

with base

~

~ ~= ~ .

~2 (p)

E Gs

.

Let Let

and

p = <0 :K < ~> ~

and

a2 (q)

2 .

~

be the

~spectively,

Since

~(~2 (p)) = 0 ,

is simple (cf. 1.5.2), it follows that Clearly

~

and

~

are not base-iso-

morphic.

Remark 1.3.12.

To complete the discussion of change of base

when one base is finite, consider the case of two with bases ble to have and

~

U,W

respectively, where

?! ~ ~

be minimal

even though Cs's

~ ~ IUI < w .

IUI ~ IWI .

with bases

Cs's

U

and

~

and

Then it is possi-

For example, let W

respectively, subject

44

Change of base

only to the condition

~ ~ IUI,IWI

1.3.13

; then

~ ~

by 2.5.30.

But

also the following simple result shows that not all possibilities can be realized

(also recall Remark 1.3.2).

Theorem 1.3.13. ated by W

Assume that

[[
such that

~ ~ ~

Proof. for each

, we have

Let

u E U

f

~

is the

Then if

~

IUI ~ IWI

Cs is any

x

= [
Crs

U

gener-

with base

.

be the given isomorphism

let

with base

9

from

Then

~

onto

~



, and

is a

U

system of pairwise disjoint elements of u E U

and

every

u E U

K,k < ~

we have

there is a

Xu ~ DKX

w E W

B .

Furthermore,

, and so

such that

for any

fXu ~ ~KX .

Hence for

<w :K < ~) E fx

Hence U

Iul

Iwl Remark 1.3.14.

of 1.3.13 Ws

By Cor. 1.4

is regular.

ofAndrgka, N~meti [AN3],

The construction

with the same conclusion as

1.3.13.

the algebra

can be modified to give a Namely,

fix

u~ E U

and let

v f

p = _
Let

~

be the

Ws

with unit element

~

~U ~p~

gener-

[p~ :u E U}

Theorem 1.3.13 and Remark 1.3.14 show that in general the size of a base cannot be reduced.

We now prove a theorem giving important

special cases in which it is possible

to decrease

the base;

the theorem

is due to Andr~ka, Monk, and N~meti.

It is an algebraic version

of the

downward LSwenheim-Skolem

theorem, and is proved as that theorem is

proved in Tarski, Vaught

[TV]

Monk

[HM]

It generalizes

Lenmm 5 in Henkin,

First we need a definition and a le=~na.

1.3.15

Change of base

Definition 1.3.15. ments

V0

and

VI

an isomorphism of to

~

, and

~

Let

~

and

respectively. ~

onto

45

9

be

Crs's

If the mapping

~ , then we say that

is ext-isomorRhic

to

with unit ele-

(X N V 0 : X E B> ~

is

is sub-isomorphic

~ .

This definition gives algebraic versions of the notions of elementary substructures and elementary extensions. isomorphic to in those of

~

~

then the unit element and base of

and

V1

Then if

Let

~

and

X E B

is regular in

Remark 1.3.17.

~

For each

isomorphic to a non-regular

~

is sub-

are contained

be

Crs's

Cs

, so is

~ ~ ~

~

with unit elements is sub-isomorphic

X N V0

in

Cs

with base 0

to

~ .

there is a regular

Cs

sub-

(thus 1.3.16 holds only in the direc-

the example is due to Andr~ka and N~meti.

, be the full

K ~ 0).

~

respectively, and assume that

tion given);

~

.

Lenmm 1.3.16. V0

Note that if

m + I , resp. (thus

qO = ~

m . and

Let

Set

~

, resp.

p =

qK = 0

for all

Set

X = [u E ~(w + I) (p) : uO = w] U {u E ~(w + I) ~ a(w + I) (p) : uO = O] .

Let

~ =~[X]

P E X , but however,

and q ~ X .

~ =~[X Hence

X N am = [u E ~

= I ; hence by 1.4.1 below, morphic to

9

.

n ~} X

Now

~X

is not regular in

: uO = O] ~

.

is regular in

is regular.

= I , ~

. ~

Finally,

llp = llq

,

Clearly, and ~

A~(X n~w) is subiso-

This follows from the following three facts:

46 (I)

Change If

~I I

and

respectively,

~'

with

of base

are simple

1.3.17

CA's 0~

generated

by

~x 0 = Ax I = I , and if for every

[Xo]

and

[Xl]

K < ~

and

~ < 2

we have

c(K)[d(K

X K) "n
--

0

c(K)[d(K

x K) "[I < K- s)xg]

then there is an isomorphism (2)

~] and

~

satisfy

(I)

f

of

~'

, with

and

I

=

onto

~'

,

such that

~ = ~i' , ~ = ~

,

fx I = x 0 .

X = xI ,

X N ~w = x 0 o (3)

If

then

f

is an isomorphism

f =
: Y E B)

It is straightforward are simple. simple all

~I

from

~(~

which generates (I)

~g~

~i

lowing statement

For

For all finite

fX = X ~ ~

into

~

f

and

agreeing

that

That both

(3) , note that

that both

,

~

f(sOx)

~

and

and ~

are for

= s0X N


are homo-

on

:K,X < ~} U Is Ox :K < ~]

by

2.2.24

(I)

and

(2) .

and

is an isomorphism

fdKk

=

R,S ~ ~ X ~

dKh

Hence

(3)

17 of Monk

for completeness.

that there

fx I = x 0

such that

(2) , except possibly

is a special case of Theorem

sketch the proof of

such that

onto

is simple by 1.5.2 below.

{D~

(4)

to check

K < ~ , and hence

Finally,

~

9

is seen by 2.2.24.

morphisms

of

for all

It follows f

of

holds.

[MI]

We

from the fol-

~$~'

onto

K,~ < ~ :

and all finite

9

F,A _c ~

,

~g~

1.3.18

Change of base

0 - dK% 9 ~KEFsKXO .~KEAS$(-Xo)

~(K,[>ERdKk "~6S iff

y

and

K ,

y = O

true by using the hypothesis tion on

cK

Assume that

Let

K

S c_ U

(i)

Given such an isomorphism

f , that

f

Crs

c~

with unit element

Then there is a

W

V

and

IAI ~ K ~ IUI .

with

S _c W = U

such

and:

Each of

is ext-isomorphic

the

following

to a

Crs

a)

9/ E Ws

b)

~ = K I~1

e)

9/

conditions

If

ext-isomorphic

is a

to a

- c)

V N

implies

that

9/

X

is

W :

; 9

is a regular

9~

a)

with unit element

Gs

, and

the least infinite cardinal such that (ii)

is

2.2.22.

9/ be a

I SI ~ K 9

(4)

2.2.22, proceeding by induc-

be an infinite cardinal such that

and

IWI = K

Let

cKy = 0 , one sees that and

is also easily seen by

Theorem 1.3.18. U .

iff (I)

IF dR U FdS U F U A 1

preserves

base

of

= 0

dK~ .~KEFSSXl .nKE s2(-Xl) = 0 .

~ERdK~ .nE S

Since for all

47

Ws

Ws

K =>

I~XI < k

, where for all

with unit element

with unit element

X E A .

~U (p) , then

9/

is

~W (p)

O~

(iii) phic to a

Cs

(iv) then

9/

9/

is a

with base If

~I

If W .

9/

Cs

and

is a

K = K I~I , then

9/

is ext-isomor-

W .

is a regular

is e x t - i s o m o r p h i c

(v) with base

If

Gs

(resp.

to a regular

Gws

then

~

Gs

Cs ) and (resp.

(i) (c)

Cs )

is ext-isomorphic

with

to a

holds, base

Gws

W .

48

Change of base

(vi) base

W

If

~

is ext-isomorphic

(ii):

Crs

We assume given w e l l - o r d e r i n g s

Note that

I~I ~ IAI

.

of

U

and

V .

There is a subset

TO

S U Rgp c T O , and

X N ~T 0 @ 0

N o w suppose that

0 < ~ < <

has been defined

Let

and let

that

IT01 = K ,

M = ~<~T

and

T

T 8 = M U [a E U : there is an u E aM(P) that

Let

such that

u~a E X}

a

for all

easy to check.

By

such that

whenever

0 ~ X E A

~M N V .

The condition

~ ~ ~

to note that

is in

~T~

for some

the conclusion

whenever

such

0 ~ X E A

for all

q

7 < $ .

U

with the property

~ < K .

is clear.

is treated by

is

We make the same construction,

T

S c T O , and

we replace

because

aM(P)

above by

N o w suppose

~

(iv):

C

If

~

(cf. I.I.8).

, a

~ E ~X

p E ~W

is discrete Then

N o w we

is defined as follows:

X E A

for

it is

~ ~ 2 , since the case

(i) (a) a n d (ii) above

= M (J [a E U : there is an

preserves

is non-discrete.

we may assume that

T8

ITs1 = K

K I~I = K , and hence any

(i) (c) and

begin-

X N ~T 0 ~ 0

is used to check that

(X N ~W ~ V : X E A>

proceed as in (i) (b), except that

ITsI = K

The desired conclusion

IT01 = K ,

K I~I = <

~ < cf~

Furthermore,

T

U

it is easily seen that

IWI = ~

; to construct

To check that

enough

~ 1

induction

(i) (b) and (iii):

TO ~ U

IAI ~ K .

of

X E A, a ~ < ~, and a

is the first element of

B ~ K ; in particular,

ning w i t h

(i) (a)

o

W = TK = ~<MT

all

with

to a

.

Proof. and

~ ~ K , then

1.3.18

, a

then

1.3.18

Change of base

E ~[~]

and a

element of

U

u

E

~X~[~U[9]~

such that

~ith the property that

( ( ~ X ~[~]) ~ [~])~ v = u

and

The condition in

(1) (c)

B ~ K , and that

u ~ FW

< K .

To check that

49

v E X

a

is the first

for some

v 6 V

with

v~ = a] .

is used to check that with

IF1 < k

h = <X N ~ W

ITsI = K

implies

for all

u ~ FT~

N v :x E A>

for some

preserves

C

,

we need to prove

(*)

c [~wnv] (x O ~w N v) = c[V]x n ~w N v

for

X E A .

If

~ ~ ~(~)X , then

Then the inclusion So

p 6 V , and

u = ((~X~

q

Pa s X

in

(*)

is clear.

is clear.

for some

[~]) U [~])Ip 9

(*)

a .

Then

Assume

Suppose

Choose

~ E ~(~)X .

P E c~V]x N ~W N V

~ s ~

[~]

Q

and let

u E (~X~[~])U[9](UT
for some

I

< K ,

so by the definition of

( ( ~ X ~ [ ~ ] ) U [9]) I v = u follows since P%~ E ~W . Thus in

~I E G s

Thus

and and

T8 ,

v 6 X

v~ E T~ = W ~ ~ 2

that

.

for some Then

v~ =

p~ v~ E V , w h i ~

,AXI v = AXI ~v~ , so by regularity of

P E c[~W~](X

N aW N V) , as desired.

(iv) , see Lemma 1.3.16.

(v):

v

with = p~ , so it

u~

obviously X '

P~v~ 6 X .

For the regularity required

Let the unit element of

~

be

U i 6 i ~ T i (pi) , where

~T~Pi J t ~ N ~T~pj~t ~. = 0 J

such that X E A .

S c_ U j E j T j Now take any

universe of a

Ws

and

for

i ~ j .

Choose

X N U j E j ~T j(pj) # 0

j E J .

The set

J c I

with

for all non-zero

IX n aT Ip3). : X 6 A] J

, as is easily checked (since

IJl K K

~

is the

may not be a 3

member of

A , we cannot use the notation

R t 91, y = ~T [pj)` ) . Y J

We

50

Change of base

denote this

Ws

by

~j

1Tj I ~ K , by (ii), ~W!pj)j , where

L6Ej~W~pj)

~j

ITjl < K , we set

is ext-isomorphic to a

rgpj U (S n Tj) < W.j ~ Tj

and

Wj = Tj .

V

such that

X n Z0 # 0

IZ01 ~ K ,

whenever

IWjl = K .

If

With

(vi):

S = ~ZEzORgZ

0 ~ X 6 A.

If

with unit element

Ws

the desired conclusion is easily checked.

be a subset of = K , and

If

1.3.19

n E 9

,

Vl =

Let

Z0

I UZEzORgZl

and

has

Zn

been defined, let

Zn+ 1 = Z < ~

Let

U [z E V : there is a

n

such that

z

V t -- Uns

q E Zn

is the least element of

and

is ext-isomorphic to a

W = ~zEviRgz Crs

.

V

an with

X E A , and a z E X

and

It is easily checked that

with unit element

VI

and base

W

9/ satis-

fying the desired conditions.

Remark 1.3.19.

The conditions in 1.3.18 are necessary for the

truth of the theorem. each

~ ~ 0~ and each infinite

finite dimensional such that

L

dition

Cs

with

of power

K I~I ~ K

Ws

I~I , with base of power

cannot be replaced by

cannot be weakened in

cannot be dropped for in

there is a locally-

Cs

o~

Cs

K I~I

with base of power in

K .

(i) a), the con-

~

K = K I~I

"Cs "

~

K

9/ is not ext-isomorphic to any

This shows that

by

First, Andr~ka and N~meti have noticed that for

(v).

Cs 's

in

(i) b) or (iii), regularity

(iv), and

"Gws "

To construct this algebra, let

cannot he replaced U = ~K 9

R = [ : f,g 6 ~U and I[K < ~ : fK ~ gK] I < w] 9

Let

1.3.19

Let

Change of base

F

be a function from

we have

Ff = Fg

iff

~U


into

E R .

51

U

such that for all

Let

~ = U

Set

X = [q E aN : qo = F(q o T)] . Clearly

Let

~

9~

be the full

Cs

with base

is locally finite-dimensional,

power

K I~I

Suppose

ext-isomorphic to any

W c U Cs

~w .

R n (=W • =W)

has cardinality

and let

IWl = K .

with base

~

and

~X = [0} .

9/ = ~ ( ~ ) [ X ]

.

W

To show that

Thus

9/ has ~

is not

it suffices to show that

Note that each equivalence class under

equivalence classes altogether. that

Ff ~ W .

Let

Then

goT

and hence

= f

U

C[oaU]x = ~U

IAI = ~ , and the base of

and

C [c~W]-(X n =w)

f,g E ~U

I~I U K < K I=I , and so there are Hence we can choose

g = [<0,f0>} U F(g o T) @ W .

Therefore

f E ~W

K I~I

such

U (~-~)I f 9 g E ~W

but

g ~ ~0 ~W] (X N ~W) , as desired. Second, Andr4ka and N4meti have constructed for each dimension-complemented

Cs=

of power

which is not ext-isomorphic to any

Cs

I=I

= a w

a regular

with base of power

with base of power

I~l+

I~I .

Thus

i

the condition

K =~

E~

cannot be dropped in

(i) c).

Third, Theorem

1.3.13 and Remark 1.3.14 show that the condition

IAI ~ K

is needed.

Fourth, the hypothesis that

K

is infinite is essential by Remark 1.3.2.

Finally, Andr4ka and Ngmeti have shown that in place

"Gs = "

by

"Gws = ".

each

= > m

that

IA! K l=I ~ IUl

9/ E Gws reg N Lf



p E ~(~)

with a base

and having the property that for all

the desired counterexample, with to define

one cannot re-

We describe their interesting example:

we construct an

IWi = Ic~l then

(i) (c)

is not an isomorphism. K = I~I.

such that for all

Let

V,6 , if

such

W c_ U , if

This provides

8 = I=21 . V < 8

U

for

< B

It is easy then

9

52

p

Change of base

~ ~ (p6)

and such that for every infinite

there is a U

6 < i

with

= m U (B~V)

Let

9/ be the

claim that

9/

~ U6

that

U

have

~ E Lf

Gws

~ ~ 6

and

V = ~

set

~ ~ 6 < ~ .

Now we show that if

Z = V n ~W ,

to any

Crs

K = IW n e I + I .

K E w 9

V

and every

For each

~U (pT) , and let

is the desired algebra. whenever

F = m

P6 E ~F .

with unit element

is not ext-isomorphic

Case I.

1.3.19

7 < ~

~ < let

X = If 6 V : f0 ~ w}

generated by

Note that

9/

IX]

.

has base

Since

5X = 1 , b y 2 . 1 . 1 5

W ~ U

and

We ~ , and (i)

we

IWI = I~I , then

9/

with unit element

V N ~W .

.

Let

We now consider two cases.

Now

c Iv] ( ~ ) , ~ r~ x ~ ) . n

<<

- s~

= v

while

so clearly Case 2.

9/

c~Z~(d(~rl--•

sOx)=o

is not ext-isomorphic

to a

K ~ ~ .

such that for all exists a

6 ,

c[V]x n Z O " 9/

Since

Crs

with unit element

IWI = 151 < cfl~21 = cf B , there is a

6 E W ,

6 < 7 .

V ~ 6 < 8 , such that

By our choice of P6 E ~(W N w) .

But we now show that P6 ~ C[oZ] (X ~ Z)

is not ext-isomorphic

otherwise there is an

to a

~ < 6

Crs

p

there then

Thus

Z ,

(which is true since

(p6)? E Z = V , using our assumptions

(p6)0 ~ Z = o% , E

In fact,

e > 6

since

Thus ~ > (P6) (p6) 0 E U 6

X , and hence

~ ~ W , by our choice of

P6 6

(p6) O E X n Z .

by the definition of

But then

~ <

and hence again

with unit element

such that

Z .

6 , contradicting

on

p ).

the fact that

1.3.19

Change of base

It remains to show that

~I

is regular.

the following fact about regularity. N4meti

(*)

~/ be a

subset of

~ .

Gws

It is a part of Lemma 1.3.4 of Andr~ka,

w i t h unit element

Suppose

that for all

(AX U F) I f = (~X O F ) I g , then

To prove

Let

For this purpose we need

[AN3], but we include its short proof for completeness.

Let

g 6 V

53

, and

Since

, and

f 6 X ,

Then

(*) , assume its hypothesis,

X

g 6 V

.

Since

fO = gO

and

k E V

and suppose

|

let

f E X ,

Finally,

[g :gg ~ g'K]

is a finite

g E X .

k' = ( ~ |

~AX

and

that

k U

is finite, we have

is a finite subset of

a finite

is regular.

f,,gt E V .

, we have

(AX U F) I f' = (&X U F) I g' , so by the h y p o t h e s i s

g' E X .

F

(AX U I) I f = (kX U I) I g ; we are to show that

[K : fg ~ f'K]

Clearly

X E A

f,g , if

g E X .

@ = F , ~ ( A X U I) , and for each

(@ x [f0])

V ,

subset of

f' 6 X . of

(*) ,

~--AX

, so

g E X , as desired. N o w we prove that we can apply 2.2.24.

~/

is regular.

Note that for any

C(~c)[~(~c x ~c) N FA~...~s~ for

Z = X

or

Z = VNX

.

Thus by

Let

Y 6 A

K 6 m

= v

.

Since

~X = I ,

we have

)

2.2.24

we can write

y = U Er[ A<~,a>~RvD~X n N~s0~D~x n ~ V

0

s~v6X

0 n C ~ 7 %v6(v~x)] ,

where

/rl

I|

< ~ , < W

6 E~

9

Let

for

R~,S~ c ~ • -r E F , a n d

and ~6,~

IR~I,IS~I E ~

< W for

for

~ E

~ 6 F ,

r

,

6 E |

,

54

Change of base

1.3.20

fl = U EF(FdR'V U FdS"V U [~%t6 : ~ E |

Note that (*)

AYc

fl .

to suppose that

g E Y 9

Since

U [',~'y~ : ~ E ~V])

Thus to prove that f E Y ,

Y

g E V , and

f E Y , choose

7 E F

9

is regular it suffices by ~ f = ~g

, and show that

so that

f E N(K,X)ER? DKX n ~(K,x~Es?--DKx 0 NNS~vsv6x ~NS~vs0 vs(v~x)

Since that

sOx = [h E V : h p ~ w} , f o r any P g E Y 9

p < ~

it

i s now e a s y to s e e

We shall consider the question of increasing bases in section 1.7, since we need ultraproducts to establish these results; see 1.7.191.7.30.

We wish to conclude this section by considering a question

related to the changing base question: ment

~W (p)

when is a

isomorphic to one with unit element

theorem is a generalization of Lemma 6 of

[H~

Ws

with unit ele-

~W(q)?

The following

due to Andr~ka and

N~meti :

Theorem 1.3.20. (the full)

Ws's

U 1 , respectively.

(i)

9/ and

(ii) and (iii) (iv)

9/l ~ ' 9/l

9/l) and VO

and

~

(resp.

~')

V 1 , and bases

be UO

Consider the following conditions:

are base-isomorphic;

there exist IUo~Rgpll

91 (resp.

with unit elements

and

~

Let

p' E V 0

= IUl'~Rgq'l

and

q~ E V I

;

;

is base-isomorphic to

~

such that

pllp~-I = qtlq'-i

1.3.21

Change of base

Then

(i) =

(ii), while

Say

V 0 = du0(P)

Proof. (iv) = ~/'

onto

then Pl

(iii):

p-I

~i

trivial. .

Choose

[p] c DK k , so = q11q

,-i

.

(ii) ~

and

(iii) =

(iii) = (iv) .

V I = ~U~ q)

(ii):

q' E V I

Let

so that

[q'} _c DKX

Also,

55

and

(i) = (iv):

f

trivial.

be an isomorphism from

f[p] = [q']

q'K = q~h 9

.

If

pK = pX ,

By symmetry,

it is easy to check that

IU--Rgp I = lid E A t : d

is an atom

C0{P} n [~O
Thus

l[d E B' : d

is an atom ~ C0[ql ] n

~OK

~ [ql] } I = I U N R g q t l

(ii)

holds.

(ii) =

sketch the proof here.

Let

(iv):

this is proved in

U0

be extended to a one-to-one function ~

with unit element

~U~ f~'p) and

If (~A)

F

Then

onto a subset of f~

from

is an isomorphism from

(U--F) I p = ( ~ N F ) I p'

[}IM] , but we

f = [ : K< ~} 9

to-one function from a subset of

Theorem 1.3.1,

"

and

U0

~/' A

U I , and it can

onto

UI .

onto the fuli

By Ws

~"

are finite sets such that

I q = (~A)

( ~ N (F U &)) I (f' o p) = (~,~ (F U A)) I q 9

f is a one-

Thus

I q~ , clearly ~"(f~l op) = ~U q) , as

desired.

Remark 1.3.21. imply

(i)

It is easy to see that in 1.3.20,

in general.

(ii)

The condition of base-isomorphism

cannot be replaced by isomorphism. theorem of Andr~ka and N~meti.

in

does not (i)

This follows from the following

56

Subalgebras

Theorem 1.3.22. with base

U

~

is a locally finite-dimensional

q E ~U , then

and

unit element

If

1.3.22

~

is isomorphic

with

Ws

~U (q)

Proof.

Let

~/ have unit element

fX = [u E ~U (q) : there is a

Using the regularity of isomorphism

to a

Ws

from

~

~

K < ~

v E X

(1.1.16),

into

~g~

~U (q) .

Now let

and

so that

~CKXI u = A CKXI v .

with

X E A

If

Ws

f

is an

with unit element

u E fCKX , choose

w 6 ~U

let

~Xl u = ~XI v]

the full

X E A .

= ~u~

For any

it is easy to see tlmt

, ~

Define

~U (p) .

v E CKX

be setting for any

k <

~ E ~x ,

W~

Lv~ Since

is finite,

~X

~CKX I w . K w a EX

.

w E ~U (p) .

Hence by regularity Now

~ ~x Now

,~CKX = ~ X

w E CKX 9

~XI WKa = ~XI u aK , so

Choose

UKa E fX .

, so a E U

Thus

~CKXI v = such that

u E CKfX 9

The

converse is straightforward.

Some results related to 1.3.20 and 1.3.22 are given in 1.7.271.7.30.

4.

Subalgebras

Our various classes of set algebras are clearly closed under the formation of subalgebras, effect.

and we shall not formulate a theorem to this

The following theorem gives an important method for forming

regular set algebras.

The proof is due to Andr~ka and N~meti.

1.4.1

Subalgebras

Theorem 1.4.1.

If

~/

is a

57

Cs

generated by a set of regular

elements with finite dimension sets, then

Proof. We shall use all finite dimensional that

B E Su ~ .

verify

(*)

f E X N Y

and

g 6 ~U .

suppose

have

fKu 6 X .

regularity

A X = ~(-X)

of

Remarks.

Then

(*)

X ,

K,k < ~ , and clearly

X N Y E B .

with

K gu E X .

Cs's

g E ~U .

be the rank of

Thus

J E J 9

since

X

and

guK E

CKX 6 B .

So, suppose

~U ,

u 6 U

g E CKX , as desired.

to this paper,

from relational

in an associated

~j

structure,

the rank of

Set

x = {{~ ~ =A : pilx ~ Ri~ :i E i} 3

Namely,

and let Oj

struc-

first-order

we can express this construction

and

we

so by the

As mentioned in the introduction

be a relational Ri

g E X

Then for some

without recourse to an auxiliary language.

j>iEI,j6 J Pi

1.4.1

F I f = F Ig ,

so

F = ~A X ~ ~CKX .

tures and the notion of satisfaction Using

In fact, we shall

Suppose

regular cylindric set algebras arise naturally

language.

B

.

K < ~ ; we show that

.AXI fKu = ~X~ g~

1.4.2.

be the set of

; it suffices to show

~XI f = ~XI g ,

and

f E CKX , and Thus

B

g E Y , as desired. X E B

To this end we verify ~XI f = ~XI g ,

~

F = ~X U ~Y ~ ~(X n Y) .

Similarly,

Finally,

Let

for all

X,Y E B ; we show that with

is regular.

DKI 6 B

- , since

is regular.

from 1.3.19.

regular elements of

Clearly

is closed under Now let

(*)

~

of regular let

~/ =

c~ ~ w .

Let

for each

i E I

and

58

Subalgebras

Clearly each member of by 1.4.1,

the

Cs

This is the same

X

structure cribed.

~

aA

generated by

X

HenCe

is regular.

described in the introduction to this paper in

terms of a language for Conversely,

is regular and finite dimensional.

of subsets of Cs

1.4.2

~ .

given any

such that

9 E Cs reg n Lf , there is a relational

~

~

obtained from

~

in the way just des-

We shall prove this, which is rather easy, in a later paper

where we discuss this correspondence

in detail.

The assumption that the dimension

sets are finite in

1.4.1 is

essential, and cannot even be replaced by the assumption that dimension complemented,

or by the weaker assumption

~

is

that the regular

elements mentioned have dimension sets with infinite complements. see this, let let

~ ~ w , let

X = { x E ~

~

be the full

: for every odd K < ~, xK ~ x0]

[K < ~ :K is odd] U I , and hence X

is c l e a r l y

regular.

But

, so

~CoX = 0

"Gws "

For any

subbases

Y

Gws's Gws and

There is a

~X = ~ Y is not;

They a l s o

in 1.4.1.

facts about

(2)

.

w , and

Clearly

&X =

Furthermore,

is a k < w such that

while

0 ~ CoX ~ 1 , s o

is not regular. Andr~ka and N4meti have shown that

(I)

with base

~ = --~x~(~)[X] E Dc

CoX = [ x E ~w : t h e r e

xK ~ X f o r e v e r y odd K < ~] CoX

Cs

To

= I

(where ~

97m~

, ~ a w ,

W

of

Gws

~

and both

established

~ ,

~[X}

,

"Cs "

cannot be replaced by

the following

interesting

is the minimal subalgebra of ~n~

is regular Iff for every two

IYI = IWI < w

~ ~ w , and

~ ):

or

IYI,IWI ~

having elements ~Y]

X,Y

are regular but

such that ~{X,Y}

I. 4.3

(3)

Suba igebra s

There is a

Gws

~ ,

regular subalgebra of ~X = I , X (4)

set that

(a)

~

(b)

if

of

~ a w , such that

Gws

~

9

~(~)X

is the largest

CoX

X E A

such that

is not regular.

the following two conditions are equivalent:

is normal ~

~n~

and there is an element

is regular, and

For every

X

~

59

(see 1.2.6);

is the full

Gws

such that

~ ~ ~

, then every sub-

consisting of regular finite-dimensional

elements is such

is regular.

We also mention the following useful and obvious property of regular

Cs's:

Theorem 1.4.3.

If

~

is a regular

Cs

For the rest of this section we consider ber of generators of set algebras,

, then

the problem of the num-

in particular,

a set algebra has a single generator. 2.1.11, 2.3.22, 2.3.23, and 2.6.25.

ZdgJ = [0,I]

conditions under which

This question was considered In particular,

in

following 2.1.11 the

following result was stated, the proof being easily obtained from the proof of 2.1.11:

(*)

If

2 ~ ~ < w

and

K < w , then the full

Cs

with base

K

is

generated by a single element.

By generalizing generalization

the proof of 2.1oll further we obtain the following of

(*) , due to Monk.

Theorem 1.4.4. unit element

V = U

Let iEl

~U

~ < • , and let i '

where

~/

U. N U. = 0 i 3

be the full

Gs

for distinct

with i,j ~ I ,

60

Suba igebra s

2 9 III ~o~

, and

1
< w

I. 4.4

for all

i E I .

Then

9/

~ ~ i ,

so assume

is generated

by a single element.

Proof. We may assume tK

The t h e o r e m that

be a one-one

generator

is trivial

I = ~ < w

if

with

function mapping

2~ ~ a . UK

onto some

~K < •

tKu2K < tKu2K+l] Now for each

K < ~

0 ~ ~ < m

n C2KC2K+IX

Y

K

E ~A

by recursion:

,

we set

Then it is easily seen by induction

on

n X) .

k

that

YKk = {u E aU K :~ ~ tKu2K+l ] k < 9 9

Hence

if

K < B

and

v E UK

we have

[u E aU K : U2K+l = v] = YK,tK v N YK,tKV+l K < ~

and

[u] = A g < ~ Thus

X

Our single

9

YK,k +I = YKk [~ C 2 K ( C 2 K + I ( Y K x N X )

Thus for

"

let

of ~ such that u E aU K , then

we define a sequence

YKO = C 2 K + I ( N X )

for all

K < ~

a ~ 2 .

is

X = {u E V : if K is the element

while for

For each

that

generates

u E C~UK

we have

s2K+l(y 9 K'tKu~-- yK,tKU +i ) . 9/ , since

A

is finite.

.

1.4.5

Suba igebras

Remark 1.4.5.

81

Note that the assumption

~ < m

in

1.4.4

is in-

essential if we replace "generated" by "completely generated". 1.4.4

has been generalized by Andr~ka and N~matl, who showed in

that if for each

~

with

2 ~ ~ < w

of pairwise disjoint sets with < ~

such that the full

Uy<$~Uy

f

[AN4]

we let

f~ = the smallest ~ such that there is a system

then

Theorem

Gs


w > IUvl > I

: y < B> for all

with unit element

is not generated by a single element,

is given by the simple arithmetic

formula

I.^2 ~ 2 2~-I f~ = ~(Z ) + I .

Also note that in Theorem 1.4.4 one may assume that sihle or even replace the assumption by the condition:

IUil < ~

most one

1.4.4

that

i E I .

U. n U. = 0 i j

is a Wj

,

that

Gs

that

for all

IUil = 0

1 < Iui I < w

i E I , and

is pos-

for all

IUil = I

i E I

for at

even remains true if we delete the assumption

for distinct

i,j 6 1 , since the assumption that

implies the existence of pairwise disjoint non-empty

j E J , such that

V = UjE/W

j , and an easy argument

shows

IJl ~ III " Closely related to

(*)

above

is the following theorem of

Stephen Comer.

Theorem 1.4.6. a base element.

U

such that

Assume that IUI ~ ~ .

~ < w , and that Then

~

~ is a

Cs

with

can be generated by a single

62

Subalgebras Proof.

x E ~U

let

We may assume that x t = ~[X

We shall show that if

~ k 2

: x E X E A} x

maps

If

x,y E ~ U

y' = CKx' N DKk

For,

,

~

, so

x E x' , and

Hence

K,k < ~ ,

onto

Clearly

U ~ 0 . x' E A

U

then

For each

, since x'

A

is finite.

generates

~

.

To

results.

K ~ k , and

y' E Sg{x']

x ~

[K/~]

= y , then

.

( ~ N [ K } ) I x = (~'~[K]) I y , so

y' c CKX' n DKk

y E X .

and

Thus

prove this we need four preliminary

(I)

1.4.6

Now let

X E A

x E CK(X n DKk)

, so

y E C K x'

be arbitrary

DKk

such that

x' ~ C K(X ~ DKk)

.

Thus

CK x ~ N DKk ~ CK(X n DKk ) N DK~ = X n D K%~ X .

CKx' n DKk c- yP

Hence

(2)

If

x,y 6 ~U

,

xK = x

= y , and

, and

K,k,~ , then

For, under the hypotheses follows

(3)

If

= y

and

(I)

is established.

are distinct ordinals

(4)

x

o [~l~,xl~]

y i E Sg~x I]

of (2) we have

y = x ~ [K/k] o [k/~] , so (2)

from (I).

x , y E ~U ,

K,k,~,~

x~ = x9 , then

are distinct ordinals

<

o~ ,

x o [KIX,~.I~]

y' = s~xsKs~u~ ~ K^ x ' n D ~

In fact, y = x o [~/K] ~ [K/k] ~ [k/~] ~ [~/9] from

< a ,

, so

(3)

follows easily

(I) .

if

x,y E ~u ,

For, write of transpositions Now let

x

Rgy ~ Rgx

y = xo~

, and

with

~ E ~

and replacements, map

~

onto

IRgx I < a , then

; express

and use

U ; we show

~

y' 6 Sg[x']

.

as a product

(1)-(3). Sg[x'] = A .

If

IuI < ~ ,

1.4.7

Subalgebras

then by (4) have

Y ' E Sg[x']

for each

63

Y E ~U , and for any

X = ~ y E X yl E Sg[x l] . Assume that

let

Ky = x o [K/0]

x = [0/I] ; again then either

; thus

(KY)' E Sg[x'}

zt E Sg[x']

Rgw = RgKy

w' E Sg~(Ky)~ ] = Sg[x'}

just discussed.

(s)

Let

by

(I) .

for some , or

Hence it suffices to take w' E Sg~x'] .

IUI = ~ .

U

t = w o [0/I] .

(I) .

Now if

and

with

K <

Also let

w E ~U

and

z IRgw I < ~ ,

(4)

w ~ E Sg[z ~} = Sg{x'] .

IRgw I = ~

Then

we

For each

K E c~N i , and hence by

Rgw = Rgz

w E

by

X E A

and show that

t' E Sg[x p]

by the situation

Thus the proof is finished as soon as we prove

w' = Cot' n ~ O < K < ~ - D0< .

Sinc e

w E Cot' n N 0 < K <

suppose

w E X E A .

~ D0K , the inclusion

Then

Cot' n N0<x<~

t E C0(X N ~ K < k < ~

= TM

is clear.

Now

DKI) , so

D0~ ~ C0(X N A~
so

in

(5)

follows.

Remark 1.4.7. I ~ ~,~ < w and

~

In contrast to

there is a

Cs

~

1.4.6

with base

we now show that for U

cannot he generated by fewer than

such that log2~

IUI = 8 "~

elements.

The exam-

ple is due to Henkln, with some simplifications by Andr~ka and N~meti. Let

~

be the full

Cs

with base

~ "6 9

For each

X~ = [q E ~(~ -8) : q is one-to-one and Z ~

Clearly

<X

:2 < B>

is a partition of

~ ( ~ X ~)

5 < 8

q

let

m 5(mod 8)] 9

into pairwise dis-

64

Subalgebras

joint non-empty sets, and K < ~ . that

Let

Y = [X

CKX~ = CKd(~ X ~)

: v < 8]

and

let

Note that

for all

9/ = ~ ( ~ ) Y

9/ cannot be generated by less than

be the minimal subalgebra of

(i)

1.4.7

log2~

9/ , and let

.

b E B

In fact, since

B

and all

K < ~ ,

Now we claim

CKb E M .

is finite it is enough to prove

CKX~ = CKd(~ X ~) Because of

(I)

for all

~ 2 IGl ~)(E that

Let [J M) .

G c A

~ =~o~(~)G 9 Since

X

(I) At~

for an atom this is clear,

~ < ~

we have, first of all,

Now suppose that

First let

~ =~0gg/)(y-- U M) .

b , and by virtue of the mentioned description of since

and

We shall show

elements.

A~8 = Y [J Ix : x E AtM, x ~ ~(c~ X r

for all

~ < ~

and

9/ = ~ ( ~ ) G

Then by

; we show that

(i) 9/ = ~ ( ~ ) ( G

c -d(~ X ~) E At-~ , and so

_ Y c [z n ~(= x ~) : z 6 At~]

B = A .

for all

U M)

~ < B

it follows

~ K IAt~l ~ 2 IGI , as desired~

It is perhaps interesting that the algebra just constructed can be generated by fact, let

7

elements, where

f 6 8(V2)

7

is the least integer

be one-to-one,

with

> log2~ .

f0 = (I : 9 < 7> 9

In

For each

< y , let

Then, as is easily checked, and for each

x Thus

[Z

5 < B

=

with

X 0 = ~9<~{ Z~ , ~(~ X ~) = CoX 0 [~ CIX 0 , 5 # 0 ,

(~(~x6)~ U[ Z ) :f~

: ~ < 7]

generates

=

0]) n ~{z :f~

9/ , as desired.

=

I] .

I .4.8

Suba igeb ra s

85

Remarks 1.4.8. G. Bergman independently obtained examples of

Cs2's

which require a large number of generators.

In connection

with 1.4.6 and 1.4.7 it is natural to define the following function. For

2 K ~,~ < w

let

q(~,~) = the smallest base

~

y < w

can be generated by

Thus the example in 1.4.7 shows that 1.4.6 says that

such that every

q(~,~) = I

for

= I ,

with

elements.

q(~,a " 6) a log2~ , and Theorem

~ ~ ~ .

Andr@ka and N4meti have

established many properties of this function q(~,~+l)

7

Cs

q .

For example,

q(2,~) = least integer e log2( 8 - i)

for

~ > 2 , and

q(~,~ "9) = least integer > Iog2(8 + ~_-~i ) ,

generalizing 1.4.6 and 1.4.7. Note that these results on the function discussion in 1.3 of change of base. if a

Cs

q

are relevant to our

For example 1.4.6 implies that

cannot be generated by a single element, then it is not even

isomorphic to a

Cs

with base of power

~ ~ .

R0 J. Larson has shown that for phism types of one-generated [EFL]

Cs 's.

show that there is a countable

generated

2 ~ ~ < w

ICs

isomor-

Cs 2

not embeddable in any finitely

Cs 2 .

and

IWs

under directed unions. ICs re~=

2

P. Erdos, V. Faber and J. Larson

In 1.7.10 and 1.7.11 we show that ),

there are

(for

IGs

~ < w ) , and

and

IGws

ICs reg N Lf

Andr~ka and N4meti have shown that

are not closed under directed unions for

~ ~ w .

(for arbitrary are closed ICs

and

It remains

68

Homomorphisms

open whether

IWs

1.5.1

is closed under directed unions for

~ m w ; the

proof of 1.7.11 may be relevant to this problem. We also should mention that Problem 2.3 of

H. Andrlka and I. N4meti have solved

[KMT] by showing that for each

finitely generated

Cs

~ > 0

there is a simple

not generated by a single element; see

[AN2].

HomomorDhisms

5.

The following result about

CA's

in general will be useful in

what follows.

Theorem 1.5.1. = {x 9

d :x E I

Proof.

and

Let

~

be a

CA

and

I s Lb~ .

Then

Sg(~)l

d E Sg~[0}}

This is clear since

~)I/I

is a minimal

CA

Turning now to set algebras, we begin with a result concerning simple algebras.

Theorem 1.5.2.

(i) Any regular locally flnite-dimensional

Cs

with non-empty base is simple. (ii) (iii)

Proof.

Any locally finite-dimensional For

a < ~

any

Cs

Trivial, using

Corollary

1.5.3.

Let

is simple.

with non-empty base is simple.

2.3.14.

~ 6 Cs

Then the minimal subalgebra of

is IuI if Iul <'n

Ws

U Ws

~I is simple.

,anditis

with base

U ,

IUl > 0 .

The characteristic of

0 if I U l ' ' n "

1.5.4

Homomorphisms

Corollary 1.5.4. e~ch

Ui ~ 0 ,

Let

I ~ 0

and

is simple iff either ~ ~ , in w h ~ h or

(2)

case

for all

characteristic

Gws's

U iEI~U i

~

be a

Gws

~ ~ I .

(I)

with subbases

Then the minimal

for all

i,j E I

has characteristic

i ~ I

we have

subalgebra of

we Pave IUil

~U i : i E I)

IUil = IUjl <

(for any

i E I ),

IUil ~ ~ ~ ~ , in which case

~

has

0 .

Remarks 1.5.5. and

~

87

.

Theorem 1.5.2

In fact, if

where

U. 0 U

subbase

UI

and let

fX = X n ~U

i ~ ~ , = 0

is non-empty,

(iii)

for

then

is a full

i ~ j

~

for all

~

does not extend to Gs

and if

is not simple.

X E A .

Gs's

with unit element

III > I

and each

For, choose

i0 6 I ,

Then it is easy to verify

i0 that

f

is a homomorphlsm of

hence that

IBI > I ), but

~

f

decomposable; For

Gs's

~ ~ ~ , 1.5.2

(i)

Also, for

with base

Gws's

.

and

So

~

Uio

(and

is not simple.

This phenomenon is more fully

Gws's

are often subdirectly

Cs's

.

does not extend to arbitrary For, let

p = <0 :K < ~>, and let

= 0 , so

~

see 1.6.3, 1.6.4.

flnite-dlmensional 2 , let

Cs

is not one-to-one.

A similar construction works for explained by the fact t ~ t

into a

~ ELf

~ =

by 2.15 (i) , while

~ ~ ~

1.5.2

(li)

~

be the full

~l(~){~2 ( p ) } ~I

locally

Cs

with base

Now

A(~2 (p))"

fails to be simple by 2.3.14.

does not extend to arbitrary

Ws's

; in

fact, one cannot even replace

"locally finite dimensional" by "dimension

complemented".

~ ~ w , and choose

N F

In fact, let

infinite.

Let

with unit element = O~z~ (~) {X}

p = <0 : K < ~>

~2 (p).

Let

Then clearly

and let

F c_ ~ ~

with

F

be the full

and Ws

X = {f E ~2 (p) :FI f _c p] , and let ~ E Ws

n Dc

while by 2.3.14

~

is

68

Homomorphisms

not simple,

since

e(A)X ~ ~2 (p)

1.5.6

for every finite

Andr~ka and N~meti have shown that 1.5.2 regular dimension-complemented

Cs's

for

that 1.5.3 does not extend to

Gs's

let

U 0 = [0]

, and put

Let

~

and

be any

Gs

U I = [1,2] (resp.

Gws

~U~ p) U ~U~ q) ); in particular, simple.

To see this, observe

homomorphic ~(IUII

image of

X IUll) ~ 0

~ , but in

~

or

we have

Remarks 1.5.6.

(ii) does not extend to

~ ~ ~ . Gws's

They also noted

for

~ m 2 .

p = (0 : K < ~) ,

) with unit element ~

that some ~

Cs

~

but it is

0

in

~

to

~

~

~

(resp. is not

U0

is a

since

Finally,

Gws

V~ = [IUil : 0 < I U i l

q =

Then

with base

is not isomorphic

In fact,

~U 0 U ~U 1

could be minimal.

1.5.4 it is worth remarking that for any

A ~ ~ .

concerning

with subbases

< ~ N ~]

(see 2.5.25).

We now discuss possible closure under homomorphisms

of our various classes of set algebras, where not all natural questions have been answered.

The situation is su~mmrized in Figure 1.5.7, which

we now discuss; of. Figure I.I.4 (I) all

~

It will be shown in 1.7.15 that

= IGs

For

the one-element

0 < ~ < m Cs

we have

IWs

) , and hence IWs

U {~] = ICs reg

(where

1

is

HCs

= HCs reg

(3)

(4)

~ ~ w 9

Andr~ka and N~meti have shown that there is an

with infinite base such that

Ws

for

U [~] = ICs reg = ICs

For the remainder of these remarks assume that

a

= HGws

.

(2)

= HWs

HGs

In 1.7.21

H~I ~ IWs

we show t h a t

U [~]

~ E Ws

o

a n y h o m o m o r p h i c image o f a

with an infinite base is isomorphic

to a

Cs

Cs

or

This is not

I. 5.6

H o m o m o r p h is ms

IGws reg = IGws

= IGs

69

= IGs re~~ = HGs

= HGs reg = HGws reg

= H~s

HCs

ICs

HCs reg

I s reg

HWs

Figure 1.5.7

true if we replace

"Cs "

"Cs reg'' , as is shown by the following

by

example of A n d r ~ k a and Ndmeti. of

~ , and let

K < ~ With

let ~

the full

that

Cs

y E A

y 6 Sg(~/)[O]

a n d let

with base

91 6 Cs reg

let

.

(and hence

(using 1.5.1).

Let

Now fix [~ N

z 6 Ig[x]

but

q~

Then

implies

that

Since

Since

such that

I~ . L~yI > ~

, as is easily checked,

We now

I ~ ~ .

z = 0

z E Sg[y]

Fix

x = UK<~y K .

I~ N A y

for some

.

Set

For each

to show that

z < C(F)X

q E C(F)Y K

"

U .

91 = ~ ) [ x ]

is regular).

Since

~ E ~ ' ~ (~z U HK U F)

~ e(F)x

that

onto

First,

d E Sg(9/)[O]

z ~ 0 .

(HK U F)] I q _c uK .

can choose

we set

z = y 9 d ; we show that

Suppose

q E z .

U

y

d

YK = C(HK)UK

We may assume

, there is a

Moreover,

be the set of finite subsets

__ reg H~ ~ IGs

but

= ~)Ig(91)[x]

as desired).

U

be a one-one function f r o m

u K = ~ X [HK]

c l a i m that regular,

H

Let

91 = ~ r y 9 d E Ig(~J){x]

, we have

y = d , l~z~Ayl

for some finite

and

contradicting

< w 9

F ~ ~ .

Thus

IA z N A y

~ s ~ ~ [K}

is

We show then

(hence

K < ~ .

~/

.

I < ~ , we

Then

k 6z qH~

z _c C(F)X

.

70

Homomorphisms

N e x t we construct by 1.4.3). Clearly N o w if

(*)

Let

y E I

then there exist

x E I , with so that

side of

(*)

in

where

(*)

side of

(*)

(5) ICs

,

example

~I

Let

y = -x

is due

and

A

.

Then

~K (p)

Clearly

I

(3)

and

, and

HWs

~ IWs

to Monk;

x ~ I

F,A

uX

~ ICs~ eg

~ = ~/I

a

,

.

and

of

-x ~ I .

such that

.

where

, but

c

y = x .

Choose

is not in the right

-x E I , w i t h H

is not

F

and

A

as

in the right

to a

or Cs

there

2 g K < w

~

HCs

is the one-

or

DK k

Ws

The

in the con-

Our c o n s t r u c t i o n

of c h a r a c t e r i s t i c

except

is

Ws

) w i t h base

ideal

We claim

of

~i .

Suppose ~

9/', ~'

.

By

> 0 , then

0 .

Let

Ws

is an error

.

p = K x [0]

Ws

subalgebras

, where

that

fact:

(resp.

or

U [I}

[De]

Let

Cs

~K,X<~

Cs

we n o w show

our example will have a finite base.

obvious

, where

Cs

(4)

in D e m a r e e

is a p r o p e r

phic m i n i m a l

(*)

H E -x , but

to

be the full

isomorphic

as in

Suppose

Then

is an a t o m l e s s

to a

subsets

uk E x

of such an example.

element

morphic

F

Of course

has no e l e m e n t

9/

, and let

~

, a eontradic~on.

Cs

~I

(hence

~ : x < z, 5 ~ nx} 9 -x))

.

on the following

If

> 2

to show that

finite

, a contradiction.

HWs a ~ ICs a

struction based

5 = Hk

In contrast

element

IZd~

, so we only need

y ~ c(r)(U6Eg(U[c6y

k < ~

with

I g ( ~ ) [ c K x . - x :K < a}

x/l E Z d ~

Suppose

(*)

I

~ E H~

1.5.6

K

I = IX : X E A, that

to the c o n t r a r y 1.5.3

(resp.

,

respectively.

~

and

~I/l that ~

unit

IXl < ~]

.

is not iso~i/I have

is isomor-

The two formulas

I .5.6

Homomorphisms

~(K • K) ~ 0 ,

hold in

~'

Noting that

d((~ + l)

and hence in ~/I

k,~ < ~ 9

Set

X

of

A

such that

X

K .

to

(*)

0 ~ X/I ~ d (~/I) k~

for

k < 5}

.

the above construction

to

HCs reg ~ ICs Andr~ka and N~meti have shown that

(8)

It is clear that

~ E HCs

clearly

has power

satisfies the above conditions.

(7)

of any

~

for exactly one

Andr4ka and N4meti have modified

show that

Gs

~ HCs

ICs reg ~ HWs

, since the minimal subalgebra

is simple or of power

~ E Gs

(9)

I

by 1.5.3, while there are

without this property.

The inclusion

It implies that

IWs

In fact, clearly

Ws

J

Hence the base of

X = Ix E ~K :xk ~ 0

It is easily checked that (6)

.

• (K + 1)) = 0

is atomless, we can obtain a contradiction

by exhibiting an element all

~J

7~

lWs HCs

c ICs reg

will be established

in

1.7.13.

, and this inclusion is easy to establish.

~ SHCs

since the full

Ws

with unit element

~

~U [p) lWs

is a homomorphlc ~ SHCs

~ HSCs

Cs

with base

It remains open whether

(II)

Andr~ka and N~meti have shown that

"Cs reg''

ICs

= HCs reg

the inclusion holds trivially if or

(12)

Hence

or

H(Cs

"Cs "

HCs

= HCs r e g .

~ Lf ) ~ I C s

is replaced by

'Us " From the definition of characteristic we know that if

has characteristic K .

U .

= HCs

(I0)

By 1.5.1,

istic

image of the full

K ,

~ > ~

, and

IBI > 1 , then

The meaning of characteristic

~

has character-

for set algebras,

described

in 1.5,.3 and 1.5.4, is further elucidated by a result of Andr~ka and N~meti according to which for each cardinal

K > 2

there is a

Cs reg ~/

72

Homomorphisms

with base of power Gws

~

in

Recall

an

element

~

l~xl < W , then

(14) a

Cs reg g

simple. base

from

has base of power

1.3.9

N

The c o n s t r u c t i o n

Now if

f E Ho(~,~)

tN N Cs

Finally,

by

(15)

for every

c Ies reg

H~ n Cs

is simple:

let

F ~ ~

and

for any

~

2.3.14

there

is

and

full

N

is not

Cs

with

Note that for any

and every

K 6 ~ N F

, then each

we have

y 6 [f[x] : x 6 ~ ]

it

By

(4)

is clear

above,

that

N

it

feIlows

is not

K

and some

Contrasting to

Figure 1.5.7

that

IN ~ I C s r e g .

K ~ 2 with

(4)

above,

there is an 1Zd~ I > 2 .

(15) , Andr4ka and N~meti have shown that

there is an IZd~l

9 6 }~/

~ 6 Cs r e g

simple.

Generalizing the construction given in

not simple, but (17)

~ cs reg

b e the

: x s Re]

~ E Cs

~ Cs r e g .

K ~ 2

, if

~ , with

, a n d so by Theorem ~.3 o f [AN3] we b.ave

with base

(16)

~ E Cs r e g

, A n d r 4 k a and N4meti h a v e c o n s t r u c t e d

Andr4ka and Nimeti have shown that for any E Cs reg

> K 9

we have

= 0 .

(*)

(13)

~ = ~)[[x]

for every finite

satisfies

such that every

~ Cs reg

to

such that

~ , and let

that

~

x , not in the minimal algebra of

In contrast

c r( )

Thus

~

F~/ N Cs

Y 6 [[x] : x E ~ }

(*)

< , having a homomorphic image

isomorphic to

(13)

1.5.6

~ 2

~ 6 Cs reg for all

with base

K

such that

~

is

~ 6 }~ 9

simplifies considerably if we restrict our-

selves to set algebras with bases and subbases infinite and to

~ ~ w 9

Let us denote by

Then we

Cs , G s ~

, etc. the corresponding classes.

1.5.8

Products

73

obtain Figure 1.5.8; see (3), (4), (9) .

Here

= ?

means that we do not

know whether equality holds in the two indicated cases,

I Cs

= HSPaoGwsof

tt Cs e g

IooCsreg

HooWs

(~ > ~) Figure 1.5.8

6.

In terms of p ~ d u e t s , Gs's of

and

Cs's ~

Products

we can express a simple relationship between

, and between

Gws

~

this relationship more generally for

's

and

Crs's

Ws's of .

.

We first express

For this purpose it

is convenient to introduce the following special notation.

Definition 1.6.1. let

W c_ V .

Then

any

a E A

rtw~a = W n a .

,

Theorem 1.6.2. ~iEiVi for all

, where i 6 1 .

r~

Let

Let

9~ be a

Crs

of

with unit element

is the function with domain

~

V i [~ Vj = 0 Assume that

be a full for

Crs

i,j 6 1

9 E CA

of

of and

A

V

and

such that for

with unit element i ~ j , and

For each

i E I

~V i = 0

let

~/. i

be

74

Products

the full

Crs~

1.6.3

with unit element

there is a unique

Vi .

f 6 Is~,Pis

)

Then

~ ~ PiEI ~'i

such that

In fact,

r2.v. = PJi o f

for each

i

iEI

.

Proof.

Clearly there is a unique

f

mapping

By 2.3.26,

each

by 0.3.6 (li).

f s Hom(~,PiE~li)

into

PiEiAi

rLvi E Ho(~,~ i)

and satisfying the final condition. i E I , so

B

Clearly

for

f

is

one-to-one and onto.

The assumption

~ E CA

Corollar~/ 1.6.3.

is not actually needed in 1.6.2.

For

~ a 2

we have

IGs

SPCs

and

IGs; eg

= Si~s reg.

Proof.

First suppose that

U i 6 1 ~U i ' where

U i n Uj = 0

for all

Let

i 6 I .

~ E CA

product

of

Clearly Cs ' s

We may assume that

Clearly

CI f

CI f

i 'j 6 1 ' and

for each

i 6 I , and let

~V i = O

for all

i E I

is an ~ omorphism of

~ ~d Pis

U i ~ U 3. = 0

i 6 1 , and again let -I

has unit element U.l ~ 0 ~, ~ , f since

~ ~ 2

~ onto a subdlrect

, as desired.

Second, suppose

for each

~

1

be as in Theorem 1.6.2; clearly and

; say

for distinct

V. = ~U. 1

~ s Gs

' each

1.6.3

a

Cs

with base

i,j 6 1 .

~, ~

be as in Theorem 1.6.2.

and ~

f

onto a

is handled by

In an entirely analogous way we obtain

Gs

.

1.1.15.

Let

U. ~ 0 .

for distinct

is an isomorphism of

The second part of

~.

V i = ~U i

1.6.4

Products

75

Corollary

1.6.4.

For

~ ~ 2

Corollary

1.6.5.

Let

~ E IGs

following

conditions

(i)

~

Proof.

,

By 1.6.3 and 1.5.2

PGws

Remarks

1.6.6. =

1.1.4,

are

For

IGws

None

1.6.8.

of set algebras

l~s

IAI > i ,

a < w

9

Then

the

;

R e m a r k 1.6.7.

Figures

= SPWs

is simple.

Corollary = IGs reg

,

IGws

are equivalent:

~ E ICs

(ii)

we have

,

~ ~ 2 and

PGws reg

1.6.4,

1.6.6

in Figure

1.6.9

= HSPGs

= SPCs

I

1.6.9

(c~ m 2)

,

extend

to

for

\

Figure

= IGs

H, S, P

W e n o w discuss

= IGs

separately).

PGs re~~

= IGws reg

under

and 1.5.8.

~ < 2

PGs

properties

summarized

= HSP~s

(treating

we have

of 1.6.3,

Closure

1.5.7,

(iii)

~ a 2

this

~ ~ I .

of our classes ; see also

figure.

= S~s

76

(I)

Products

For

d ~ 1

1.6.9

the diagram is different;

then the classes are just

five in number, increasing under inclusion:

(2)

[~ E CA

IWs d

(b)

ICs~ = [~ E CAd :~ is simple or IAI = I] ,"

(c)

PCs d

(d)

HPCs

(e)

HSPCs

=

[~ E CAd :~ is a product of simple CAd

's] ;

;

d d

The example

general, for

:~ is simple]

~

(a)

= CA

9/

=

d

SPWs

d

in 1.5.6 (5) also shows that

d ~ w 9

In fact,

(*)

HWs

~ PCs

d

continues to hold for all

in 9/ E PCs d

(3)

Andr~ka and N~meti have shown that

for

d ~ w 9

(4)

To show that

with bases

K,X

PWs

~ HCs

d

c~

for

Cs reg ~ PWs

d > w , let

respectively, where

~

and

and

I < K < k < w 9

since each non-trivial homomorphic image of a

Cs

~

Then

Cs

be

~ PCs reg

Ws's 01

~ X ~ ~ HCs

has a well-deflned d

characteristic, while

~ • ~ does not (cf. 2.4.61 for the definition of

characteristic). (5)

For any

d

we have

SPCs

d

~ HPCs

d

(for

d ~ w

this was shown by

01 '

1.6.9

Products

A n d r ~ k a and N~meti). while

if

For

9/ 6 H P C s

~ = 0

and

For

let

be the subalgebra of

elements.

Suppose

trivial

Cs

must have

0-

w i t h base

where let

~

.

Then

~

with base ~

~i

elements,

For each

But

~ E 9 N 2 ,

Let

a non-

IZ~I = m

1.5.2 PiE~j

we

(iii), it has only

a contradiction. let

and let

For, assume

i E I .

m , and

~.~

since

is simple by j c I .

,

zero-dimensional

result of Koppelberg,

~ HPCs

for all

Cs

= BA

by S. K o p p e l b e r g

PiEI ~ i > ~) , each

~ = P56~,~ 297~

~/

~

be the minimal sub-

that

~D = PiEl~i

be the minimal

h 6 Ho(PiEl~i,~ ) ,

.

For each

5 s ~ N 2

be the term

e(~)d(5• Thus

; say

(SPCs

IAI m = IAI

generated by

for some

~ > ~ .

~ , let

~i 6 Cs ~

~

dimensional

N o w suppose

then

be the full

Since each

~ = PiEj~j

finitely many

algebra of

~

By the above

follows that

Cs

let

~ E HPCs~

III < ~ .

this is w e l l - k n o w n

IAI > 0u

[Ko]). 9

0 < ~ < 0~

77

@

= 1

It follows there is an f 6 Pi61Ci

holds

that

i 6 1

in a

Cs

~ 0

~

~ E w N

is even, and

~i

fi = 1 1 ,

~

~(~9~)--" h f = 0

so

= 0 , so by 2.1.17

has base of c a r d i n a l i t y

has base of power if the base of (~i)

~(~) 9 f = 0

and

.

~ E ~ ~ 2 , so for all

(~i)

fi = 0

~(~)-- ~ hf A(~)hf

iff

for all

such that

b y defining

ments for some E w'~ 2

~3) &

~) .- c(5+l)d((5 + I) • (5 + I))

otherwise.

if

Now define

has

Thus

But

hf E S g ( ~ ) [ ~ ( ~ )

2~ ~

~ E w N 2 is odd.

in these two cases. (Ill),

~ E ~ ,~ 2

~ . ~i

~ .

(~)

ele~ f

Hence

~(~D)f -- 0 , :~6~2]

w h l e h is c l e a r l y impossible. (6)

From

(4)

it follows,

of course,

that

Pes

~ ICs

if

even for

,

78

Products

~ ~ bases

(7)

.

B u t we show in

is isomorphic

to a

Cs

that a product for

~ k w

A n d r ~ l ~ and N d m e t i h a v e n o t e d t h a t

for all

~ > 0 .

and

~

let

Then by

(*)

of

Cor.

1.4

similar: (8)

of

x

Dc

(*)

a n y full

.

Then

(**)

for all

x , if

holds

in every

Lf

infinite

Cs

and

Ws

w i t h base

by the atoms

N SPDc K ~ 2 ,

of

~

.

But the statement

k < ~

in every

(7)

we

then

x = 0

~ E SPDc

~ ~ SPDc

Since

The case of

Dc's

should m e n t i o n

not

In fact, write

IFI,II

full

9 E Cs r e g .

for all

with

every a t o m Ws's

is

has an atom.

with and

Cs's

Cs r e g ~ SPDc

generated

, we have

Ws

Lf's

~ k ~

,

~

of

.

be the

of

cBx = 0

falsifies

fact about

N

, and hence

In c o n n e c t i o n

for

let

JAN3]

, if

in every ~

In fact,

be the subalgebra

for all

holds x

1.7.21

1.6.9

found

= F U g

in

the following

[HMT] :

with

SPDc

F ~ & = 0

general

~ SPLf and

the s t a t e m e n t

c~x = 0

, hence

for all

in every

k E F

SPLf

then

, but

x = 0

fails

in some

Dc

c~ (9)

F r o m 1.6.13

(I0)

Among

important (II)

and subbases use

the q u e s t i o n s

seems

If we

it follows

about

to be w h e t h e r

restrict

1.6.9

of 1.5.7

PWs

Figure

~ IWs 1.6.9 w h i c h are open the m o s t

ICs reg ~ HPWs

ourselves

infinite,

the n o t a t i o n

that

to

~ > ~

simplifies

(17)

.

and

to set algebras

as in F i g u r e

1.6.10,

w i t h bases w h e r e we

1.6.11

Products

79

l=oCs~ = HSP Gws =? reg HP C s _ _

H Cs

re

Ws

tiP

"~ csreg

] ~

I

W

s

(~

> w)

Figure 1.6.10 Again

= ?

means that equality of the classes in question is not known.

Some of the theorems needed to check this figure are in

[AN316,2.

Now we discuss direct indecomposability, subdirect indecomposability, and weak subdirect indecomposability.

We give some simple results

about these notions and then we discuss some examples and problems.

Theorem 1.6.11.

Proof.

Let

0 ~ y E A , choose (~

~

Every full

be a full f E y 9

F) I f = (~ N F)I p .

Ws

Ws

is subdirectly indecomposable.

, with unit element

Then there is a finite

Thus

[p} ~ C(F)y .

So

~U (p) .

F = ~ ~

Given

such that

is subdirectly

indecomposable by 2.4.44.

Corollary 1.6.12. phic to a

Ws

Any subdirectly indecomposab~e

Cs

is isomor-

80

Products

Proof.

By I.I.II and 1.6.4.

Corollary 1.6.13.

Proof.

1.6.13

Every

Ws

is weakly subdirectly indecomposable.

By 0.3.58 (ii), 2.4.47 (i), and 1.6.11.

Corollary 1.6.14.

Let

Then the following two condi-

~ E IGws

tions are equivalent: (i) (ii)

~ E lWs ~ = ~

Proof.

; for some subdlrectly indeeomposable

(i) implies (ii) by 1.6.11.

Corollary 1.6.15.

Any regular

Cs

9 6 IGws

(ii) implies (i) by 1.6.2.

with non-empty base is directly

indecomposable.

Proof. By 1.4.3 and 2.4.14.

Remarks 1.6.16. (1)

Throughout these remarks let

Examples (I) and (II) in 2.4.50 are

~ ~ ~ .

Ws's

which are res-

peetively subdirectly indecomposable but not simple, and weakly subdirectly indecomposable but not subdirectly indecomposable. (2)

To supplement our discussion of homomorphisms we shall now

show that for any

K ~ 2

there is a

morphie image not isomorphic to a

Ws

Ws

with base

K

having a homo-

The first such example was

due to Monk; the present simpler example is due to Andr~ka and N4meti. Let

p = <0 : K < ~>

~K (p) fK # 0

Let

and let

~

x = [f 6 V : ~ ] f ~ p

is even}

Let

be the full

Ws

or the greatest

with unit element K <w

such that

1.6.13

Products

i = ig<9/)[o(r)x

We claim that

F ~ ~

(~K)I all

and

P] 9

f s y

F ~ K 6 ~ , then Thus for every

~/I ~ Ws

by

1.6.13 and 0.3.58.

Y 6 I

there is a P 9

K < W

Hence,

isomorphic

to a

iff

9/

(4)

Cs

x,-x ~ I .

is isomorphic

The full

Cs

images

that any direct factor of a

Note that a

CA

9/

to a compressed

f =

such that for

In contrast to the situation for homomorphic 1.7.17

x,-x ~ I .

c(F)x "-x ~ If 6 V : ( ~ N K ) I

(w ~ K ) I f = ( ~ ) I

1.5.6(5)), we show in

Cs

Irl < |

x/l E Zd(9//l) , so it suffices to show that

we have

(3)

.-x :r ~ |

IZd(9//l)l > 2 , hence

In fact, clearly If

81

(see

Cs

is

is a direct factor of some Gws

(of. 1.2.6).

with base of any cardinality

~ 2

is directly

decomposable. (5)

A

Cs reg

which is directly indecomposable

not weakly subdlrectly Example

indecomposable

(III) of 2.4.50.

and let

9/ be the

Cs

Namely,

(by 1.6.15) but

can be obtained by modifying

let

of subsets of

p = C0 : K < ~> , ~2

q = ,

generated by

[[p],[q}]

01

By Cor 1.4

of

[AN3],

9/

is regular, and it is clearly not weakly

subdirectly indecomposable. (6) a the

Example

Cs reg 9/ which is subdlrectly Cs

of subsets of

Clearly

9/

Now let

0 ~ x 6 A

and

(I) in 2.4.50 can be similarly modified to yield

~2

generated by

is not simple, while

d 6 Sg(91)[O]

[p} c C(F)X .

. .

Hence

indecomposable

By 1.5.1

9/ E Cs reg c~ write

but not simple:

[[p]],

is subdirectly

p

by Cor. 1,4

x = y 9 d ,where

Clearly then there is a finite 9/

with

is

as above. of

[AN3]

y E Ig(9/)[[p]]

F c ~

indecomposable.

9/

such that

82

Products

(7)

Example

1.6.13

(II) in 2.4.50

w h i c h is w e a k l y subdirectly

can be modified

indecomposable

to yield a

Cs

but not subdirectly

indecom-

posable.

This was noted by A n d r l k a and N~meti, who also constructed

a

with these properties.

Cs reg

the full

Cs

with base

2 .

To construct

Let



wise disjoint infinite subsets of each

K < ~

such a

~ .

Let

Cs

p = <0 : K < ~>

, and for

Let

9

Thus by

I = Ig(~)[X K : K < w]

we have

(*)

Now

be

let

'~ = ~ '(9 / ) [ X K : K < w]

1.5.1

~

be a system of pair-

X K = [f E ~2 (p) :F K I f = F K I P]

Let

, let

B = Ix 9 d : x E I

~

is not subdirectly

2.4.50 with

(II). x 6 1

finite

and

~ c c~

p 6 c(|

that

d E Sg(~)[O}

such that

x c

(*)

write

Then we can choose

Uh
c(~)X

| _c ~ .

indecomposable.

x 9

~ 0 .

d 9

~k 6 F k ~

Since

For any

p E c(|

Case 2.

by the same a r g u m e n t as in

0 ~ y 6 B ; by 9

.

.

K 6 ~

y = x ~ d and a

We shall show that

Since this is then true for every

y E B , it then easily follows from 2.4.46 that ~

subdireotly Case I.

d E Sg(~)[0]]

indecomposable,

for some finite

non-zero

hence

N o w suppose

and

f E x .-d

(~ U ~d)

.

we have

f E d 9 -x .

Let

~ = [~k : k < K]

k ~ x , but

.

f E ~2(P)

, and

| c ~ .

Fix

k = [ ( ~ N ~ ) I h] U similarly,

y ~ 0 , we have two possibilities.

for some finite # 0 .

is w e a k l y

For all

Then

h 6 d

k 6 ~2 (p) .

Hence

,

~ < K

choose

h = [(Ad) I f] U [ (~'~Ad) I p] since

d

is regular,

k s d

the desired c o n c l u s i o n follows

1.6.13 as

Products

in Case

I.

(8) 1.6.15 to a

Andrdka

fails:

w i t h base

and N 4 m e t i have

there

Cs r e g .

Let

K .

is a directly K a 2

Choose

F ~ ~

p = <0 : K < ~}

.

and

qo ~ ~K (ql)

For each

Let

~ = ~)[Y'Xo'Xl]--~r that

~

struction.

Let

1.4.1.

let

Now

Choose

c = [a E A

"

Note

that

and

E = [l~
(*)

C

A : E

To prove Clearly

~

be the full

0 E F

, and b o t h

qO,ql

E ~K

so that

F, ~ F

Cs infinite.

F I qo = r I ql = r l p

= If E ~K(q~) : F 1 f = r l p]

We claim

that

~

is the desired

indecomposable,

-a

Thus

is

| ~ ~]

~

algebra.

we need an a u x i l i a r y

is a regular

for some

~ c(@)x

Cs

con-

, by T h e o r e m

5 < 2

and

.

=

Let

: k < ~, a E kD}

-

D = [~
:K < w, a 6 K ( C U B ) }

N o w we claim:

.

, since

is closed

Y , X o , X 1 E E , it suffices under

Since

C U B

is closed

closed

under

c K , where

for all

let

not isomorphic

x

is closed u n d e r

(*) E

and

Cs

wlth

.

or

some finite

indecomposable

of T h e o r e m

let

= ~)[y]

:a

that the converse

5 < 2

is directly ~

shown

be arbitrary,

Let

To check

83

Z 6 D .

for some f i n i t e

cK ~~

Thus

+

, and

under

let

, so is

K < ~

a ~ ~ cKa ~ ~

E

and

for all

.

, it suffices

k < w

| c_ ~ , some

dKk 6 E

to s h o w that

To show to prove

a 6 k(C U B)

5 < 2 , and

c(~ u { ~ ] ) x

some

.

E E SugJ .

K,k < ~ that

E

that If

9 is

CKZ E E a

~ < k , then

<

c (e)x

.

84

and so

Products

c K ~
E C ~ E .

~
where and an

b E B ,

Thus we may assume

that

= ~ <~pd~ 9 b ,

~ < ~

~p < 2

1.6.13

, and for each

such that

p < ~

-dp ~ c ( |

there is a finite Let

z = ~
.

Then

. d ) 9 CKZ = (c~n < d ) 9 c b = (H
(c~n

and for each

~ < ~

we have

c~(-d ) ~ Ce~u[~])x and hence

(**)

-c~d

(c~n

E C o

Thus we conclude

_ d ) 9c z E E

g

On the other hand,

-cKH<(pd

and for each

9 C Kz = I

~ < ~

~
9 CKZ]

we have

e (-d).cz ~ c(| Thus proved

-c~

<~0d~ 9 c z E E , so by

(*) .

z 6 [0,I]

Now let

by 1.6.15.

(**)

z E Zd~ By

,

cKz E E .

Hence we have

We shall show that

(*)

z E B, hence

we can write

z =)z~ 9
where

K < ~ ,

is a fSnite

b E KB ,

subset of

~

-d and

c(|

~ ~

< 2 ,

for each e ~ c(n)x 0

~ < K , where and

|

g ~ x(E)x I ,

1.6.13

Products

where

Q

and

~

are finite

85

subsets

of

~ .

Choose distinct

X,~ E r ~ (U,,,,,,~Ab v U U v ~ @ v U ~ U E) 9 Note

that

c~ - c ( |

= 1

z = c z =)~

Furthermore,

b

for each

+ c e + c g

9 < K .

Hence

e

cBck~c (~) oX = 0 = cBkc c(E)x I , SG

z = c~kc z = I

~
as desired. It remains where

~

< 2 . for

is a Since

(9)

for all

is isomorphic

Cs

from

u 6 U

f E x 0' 9

y' = Fy

and

f0 = u

,

x' = Fx

for all

x 0 ~ y ; it follows for

for

x~

.

f E y'

that

fK = u

Since

that any subdirectly

indecomposable

Cs .

problems

about these notions

indecomposable

Cs

renmin open: isomorphic

to a

?

to a

Ws

?

hold

cannot be regular.

I s every weakly s u b d i r e c t l y indecomposable

isomorphic

F 6 Is(~,~)

= 0 , similar equations

Similarly

~

1.7.13

subdirectly

Suppose

such that

K E F , an4

to a regular

The following

Let

0 y 9 sly 9

x 0 "x{ = 0 ,

Is every weakly

regular

U .

l

and

It follows

(I0)

(b)

and

K E P , for all

~ x 6 = ~x~ = F

~ ~ ICs r e g .

with base

COY = I

x 0 ~ d0K

for all

(a)

Cs

y', and so there is a

Also,

Cs

to show that

Gws

(or

Cs reg)

88

Ultraproducts

7.

1.7.1

Ultraproducts

We shall begin with a general lemma (due to Monk) about ultraproducts of

Crs's

.

To formulate this lemma it is convenient to intro-

duce some special notation.

Definition 1.7.1. U =
y E PiEIUi/F (ii) mapping q E

PiEIUi

we have c

~

I ,

an ordinal.

such that for all

K < ~

c

mapping

and all

c(K,y) E y 9

is an

~(PiEIUi/F)

(PiEiUi/F)

be an ultrafilter on a set

(F,U,~) -choice function we mean a function into

If

F

a system of sets, and

By an

X (PiEIUi/F)

Let

(F,U,~)- choice function, then we define

into

and all

PiEI Ui

+ c

by setting, for all

i E I

+ (c q)i = . (iii) Ai _c Sh(~Ui ) tion.

Let

A =

for all

be a system of sets such that

i ~ I , and let

Then there is a unique function

c

be an

(F,U,~)- choice func-

Rep(F,U,~,A,c)

(usually abbrevi-

ated by omitting one or more of the five arguments) mapping into

Sb~(PiEIUi/~ )

such that, for any

Rep(a/F) = [q E

Lemma 1.7.2.

Let

a E PiEIAi ,

(PiEIUi/F) : [i E I : (c q)i E ai} E F] .

F

be an ultrafilter on a set

a system of non-empty sets, and choice function.

Pi61Ai/F

Further, let

~

an ordinal.

~ E Icrs

Let

where each

I , c

U =
be an ~i

(F,U,~) -

has base

Ui

1.7.2

Ultraproducts

and unit element Then

Rep(c)

Furthermore,

for all

that if

c'

f

E < ~ , then

f = Rep(c)

preserves

+ .

Now let

Kk

c T

PiEI~/i/F , if

into a

function

Crs

Z E F ,

s E PiEiVi

it follows

such that

cS(K,wK/F)

=

Rep(c') (a/F) # O.

Let

fd

.

w = <(siK : i E I> : K < ~

(F,U,~)- choice

Proof.

Note that

from

0 # a/F E PiEIAi/~

i E Z , and

is any

for all

V =

is a homomorphism

for every

si E a i

wK

V i , and set

87

,

X = PiEiUi/~

,

T = f(V/F)

K,k < ~ ; we show that

since

f

preserves

+ .

f

.

Clearly

preserves

d

q E T .

Then

Now let

Kk

+ [i E I : (e q)i 6 Vi} E F , and so

+ q E fdKx

iff

+ = (c q)i k] E F qK = q%

Now let

iff

a 6 PiEiAi

f(-a/F) c T ~ f(a/F)

Iv i]

[i 6 I : (e q)i 6 DKk iff q E

9

iff

[i 6 I : (c q)i K

[i E I : c(K,qK) i = c(K,qk)i]

E F

iff

D [T] Kk

We show that .

+ ] E F

Now let

f

preserves

q E T ~ f(a/~)

.

Clearly

Thus

[i E I : (e+q)i E Vii 6 F and [i s I : (c+q) i E ai] ~ F , i.e., + [i 6 1 : (c q)i 6 V i ~ ai} E F o Therefore q E f(-a/F) . Next,

let also

K < ~ ; we show that

f

preserves

cK .

First

+

suppose M E F . all

that

q E f(cKa/F)

Then there

i E M .

is an

We show that

[i 6 1 : si = c(K,s/F)i] any

i 6 Z N M

we have

.

9

Let

IV i ] M = {i E I : (c q)i E C K ai];

s 6 PiEiUi qK Then

such that

E f(a/F) Z 6 F

.

since

Let

[ (c+q)i ]Ksi 6 a i

thus for

Z =

e(K,s/F)

s s/~

.

For

,

88

Ultraproducts

-)i (c+qKs/F

1.7.3

= -- [(c+q)i ]~c(K,s/F)i = [ (c+ q)i ]~si E a i

Thus, indeed,

qK

s f(a/F)

Hence

q 6 c[T]f(a/~)

9

Second, suppose

s/F that

q 6 c[T]f(a/F)

9

Thus

q E T

and

(~ ~ [K}~I q = (~ N [K] J p

for

+ some P E f(a/F) . Let M = [i E I : (c P)i E a i] ; thus M E F . Also + since q 6 T the set Z = [i E I : (c q)i E Vii is in F . Now let + + + i E M N Z . Then (c q)i 6 V i and (~'~ [K])I (c q)i - (c P)i E a i , proving that

(c+ q)i E C K[V i] a i .

We have now verified that

Thus f

q 6 f(cKa/~) , since

is a homomorphism.

M N Z E F .

For the second

part of the conclusion of the lemma, assume its additional hypotheses. Let

q = <wK/F :K < ~>

(hence

and

fl = Rep(c') . We show that

f(a/F) # 0 , as desired).

In fact, for any

i E Z

q 6 f(a/F) we have

+

= <(wK) i :K < ~> = s i s a i 9

So

q E f(a/~) , as desired .

Next, we derive some specialized versions of Lemma 1.7.2

which

are more easily applicahle~ they are due to Andr4ka and N~meti.

Lemmm 1.7.3. that

F

is

(i)

Assume the hypotheses of Lemma 1.7.2.

I~I + - complete (which holds, e.g., if for any

(F,U,~) -choice function

Rep(c I) ; (ii) (iii)

Rep(c) For any

is an isomorphism; a 6 PiEIAi

we have

c'

~ < w).

we have

Also suppose Then Rep(c) =

1.7.4

Ultraproducts

89

(Rep(c))(a/F) = [q 6 ~(PiEiUi/~) : there is w 6 ~(PiEIUi ) such that

qK = wK/F for all

K < ~

Proof. It suffices to prove arbitrary then implies So, let

a 6 Pi6iAi

and

q E

s ai] E F] .

(iii), since the fact that

(i), and thus

and

[i E I :PJ i ~

(ii)

c

is

follows from Lemma 1.7.2.

(Pi61Ui/F) .

We want to show that the

following two conditions are equivalent:

(I)

+ [i E I : (c q)i 6 a i] E F

(2)

there is a

and

[i E I :PJi ~

If

(I)

w E ~(PiEiUi )

(2)

w = ; then

holds, and let wK

and

K < ~ , both

c(K,qK)

set

Z K = [i E I : (wK) i = c(K,qK)i] is also in

F .

(2)

H = [i E I :PJi ~

any

Y = H N ~K~. ZK

for all

qK = wK/~

are members of is in

For any

F . i E Y

is clear.

E a i] -

(I)

Therefore the set we have

Assume the hypotheses of Lemma 1.7.2.

Nee (c)*(Pi61~i/F) . ~/ E IK

latter case, (ii) 6 K

for

~

Then for

6 ai ,

holds.

Lemma 1.7.4.

(i)

Now

qK , and hence the

+ (c q)i = = <(wK) i :K < ~> = p j i ~

and hence

K <

6 ai] 6 F .

holds, we let

suppose that

such that

~ =

Then:

implies has base

If in addition K = Gs

Let

or

9 E K

for

K = Gws

or

K = Cs

; in the

PiEIUi/F . F

is

K = Ws

I~I+ - complete, then

9/ 6 IK

implies

90

Ultraproducts

Proof. ~U. i

for each

X = PiEIUi/F so

Let

i s I . .

Let

q E f(V/~)

with base

f = Rep(c)

.

First suppose

~ E Ics

We need to prove that

q E ~X .

Then

Thus

V. =

f(V/F) = ~X , where

(c+ q)i s ~U.i = V.l

for all

The converse being obvious, we thus have

i 6 1 '

~ E Cs

X .

Next, suppose that

vi where

.

1.7.4

:

~ E IGws

U[~y!pi0): j 13

~y(pij) n ~v(pik ) = 0 ij -ik

Now for each

Then for each

i E I

we can write

Ji ] ,

whenever

j # k .

Let

S.. = ~v~PiJ'_..(~ 13 13

we set

j E PiEiJi

+ Wj = [ q E ~ X : [ i E l : ( c y Qj = PiEl

q)i E Si,ji} E F],

../~(U) i,jl

Now we claim

(I)

f(V/F) = U [ W j : j E PiEiJi ] 9

For, let Choose

q E f(V/F)

j E PiEiJi

.

so t h a t

q s Wj , as desired.

(2)

If

Let

j,k 6 PiEiJi

+ M = [i E I : (c q)i E Vii +

,

(c q ) i 6 S i , j i

i 6 M .

Clearly each

and

for all

W.l ~ f(V/F)

j/~ # k/~ , then

so that

, so (I)

M E F .

Thus

holds.

Wj ~ W k = 0 .

For assume the hypothesis of (2) and let q E W . . Let Z = 3 + + [i E I : (c q)i 6 Si,ji ] and let H = [i E I : (c q)i 6 Si,ki] 9

Then

Z n H c [i E I : ji = ki] ~ F , while

q ~ Wk ,

as desired.

Z s F , so

H ~ F .

Thus

1.7.4

Ultraproducts

(3)

For any

j E PiEiJi

For, first let and hence

we have

q E W.. 3

Let

W.j = ~ ~ q E W j ~^(q) ~j

K < ~ 9

+ [i E I : (c q)i E Si,ji] E F

Now

[i E I : c(K,qK) i E Yi,ji ] E F , and further, since

c(K,qK) E qK ,

qK E Qj

Thus

tion, it suffices to take K E W.. qu 3 E Yi ji } "

Then

e(K,u) E u .

~

in

q E Wj ,

(3)

holds.

K < ~ ,

+ M = [i E I : (c q)i E Si,ji]

Let

,

91

M E F

So

since

M N Z E F .

q E W. J

Let

u E Qj , and show that and

and

For the other direc-

Z = ~i E I : C(K,u) i

Z E F

i E M N Z .

since

u E Qj

and

Then

+K (c qu)i = c(K,u)i + = [(c q)i]c(~,u)i E Si,ji 9 Thus

q~ E Wj , as desired. Now

that

(I), (2), (3)

Wj = W k

if

immediately yield that

Let

~ E IGs

Since

(ii) . Gs

notation, where in addition for each Yij = Yik ~i

(4)

or

is a normal

If

Yij n Yik = 0 Gws ).

j,k E PiEiJi

In fact, assume

So, assume that

= Gws i E I

F

is

I~I + -

, we can assume the above and

j,k 6 J. l

we have

(that is, in the terminology of 1.2.6,

We claim

and

j/~ ~ k/F , then

y E PiEiYi,ji 9

H ~ ~i E I : ji = ki] ~ F , so holds.

, upon noting

j/F = k/F .

Now we turn to the proof of complete.

~ E Gws

Let

H ~ F

Qj ~ Qk = 0 .

H = [i E I :Y i E Yi,ki ] and hence

Y ~ Qk "

So

Clearly (4)

92

Ultraproducts

By

(5)

(I), (2), (3)

For any

To prove

j E PiEiJi

K < ~ .

(4)

it now suffices to show

we have

q E ~ Qj .

(5) , let

for all

and

~Qj _c f(V/F)

i E I

we have

Hence by Lemma 1.7.3 we conclude that It remains to check E lws

Since

each

Ji

Pi,ti

'

Pi61Ji

Ws

~ Gws

a singleton W

.

and

Q

Thus by

(ii)

[ti}

for (i)

for

Wj

and

Qj

(6)

For any

y E ~(PiEIYi ) Let

q E W

iff

, as desired.)

so that

So, we suppose that

Y.l

for

where

j

ZK E F

(3) ,

for

W = U q 6 w ~ Q (q) Now we claim:

~s

(6) , let

for all

K < ~ 9

For each

K < a

iff

Pi

is the unique member of

r 6 ~X .

To prove

qK = yK/F

Thus

Thus

Y.1,ti '

[K < a : qK # rK]

G = [K < ~ : qK # rK] .

[i 6 1 : YKi = pi K} 9

.

K = Ws

," we write

r = ~/~ :K < ~> 9

W = ~Q(r)

qK = yK/~

, we still have the above situation, with

Let

(Hence

and

PJi ~ y 6 ~Yi,ji ~ Vi "

q E f(V/F)

f(V/~) = W , and by

q 6 ~X ,

.

y E ~(PiEiYi ,ji)

Say

Then for all

1.7.4

K ~ G .

finite.

q 6 ~X .

let

Choose

ZK =

Hence the set

Z = N [ Z K : K s a'~ G] [~ N I l N Z K : K E G]

is in

F

(by

I~I + -completeness).

[K :YKi = pi K] = ~ ~ G . M 6 F

i E Z

we have

M = [i 6 1 :c(K,qK) i = YK i] 9

Thus

also.

Now if in

Let

Now for all

F .

For

c(K,qK) i = yK i is finite.

q s W , then the set i s M n N n Z

N = [i E I : (c+q)i E ~y~pl)]

we have, for all

and hence (by the definition of

K < ~ , N),

is

(c+q)i K =

G = [K :YK i ~ PiK}

1.7.5

Ultraproducts

On the other hand, any

K E ~ -- G

if

we have

G

93

is finite then for any

(c+q)i K = c(K,qK)i = YKi = pi K

+ ~y(pi), [i E I : (c q)i E i J 6 F

M n Z E F ,

i E M N Z

and

and

and so, since

q E W .

This completes

the proof.

Lemma 1.7.5. or

Gs

Assume the hypotheses

Then for every non-zero

choice function

Proof.

c'

If

such that

F

is

of Lemma 1.7.2.

a 6 PiEIAi/F

Rep(c')a # 0

I~I + - c o m p l e t e ,

Let

K = Ws

there is an

and

(F,U,~) -

Rep(cl)*(Pi61~i/F)

the desired conclusion

from Lemma 1.7.3 and Lemma 1.7.4 (ii). So we assume henceforth is not

I~I + - complete.

In particular,

~ ~ w .

as in the last part of the proof of Lemma 1.7.2 Assume now

~ 6 ~s

Let

easily seen that there is an

We let .

Let

s

(F,U,~)- choice function

follows that

and

w

X = PiEIUi/F

N = [i E I : IUil > I] .

E K .

F be .

Then it is

c'

satisfying

the following two conditions:

(I)

ct(K,wK/F) = wK

(2)

c'(K,y) i # wKi

Again let <wK/F:K q E~X

for all

K < ~ ;

whenever

K < ~ ,

f = Rep(c t) . < ~>

.

Y E X ,

By Lenmm 1.7.2,

We shall show that

y # wK/F

fa # 0 .

, and

Now let

f(V/F) = ~X (q) .

i E N .

q =

Note that

. If

N ~ F , then

~x (q) , so

(ct+P)iK = siK

as desired.

and hence

0 # f(V/F) ~ ~X = [q] =

f(v/~) = ~x (q)

Assume that

p E ~X (q) 9

IXl = I

iff

Since

N 6 F .

Let

y/~ = wK/F N E F

P E r

,

, and hence

it follows that

i E N , and

K < a 9

(c'+P)i E V i p 6 f(V/F)

iff

Then

iff p E ~X (q)

94

Ultraproducts

Next, assume is an

h E I

of

V.

l

Since

~ E IGs

such that

l

choice function

cp

is not

I~I + - complete, there

(see, e.g., Chang-Keisler

i E I

there is a subbase

Now there clearly is an

(F,U,~) -

l

satisfying the following three conditions:

(3)

c1(hi,y/~) i E Yi

(4)

et(K,Y/~) E PiEIYi

(5)

c~(K,wK/~) = wK

Let

W = PiEIYi/F (U)

for all

and all

K < ~

and all

f = Rep(e')

f(V/~) = ~W . that

Y E X ; Y E PiEIYi ;

~ < ~ .

Again let

o

such

i E I

for all

for all

Now we shall show that q E ~

F

, for each

s. E ~Y. 9

l

Now l e t

Since

[I/(h Ih-l)] N F = 0

such that

[CK], p. 180). Y.

~ E IGs~

1.7.6

qK ~ W

H = [i E I : c(K,qK) i ~ Yi] E F .

If

By

for

(4)

some

By Lemma 1 . 7 , 2 ~

.

we have K < c~ .

i E H , then by

fa ~ 0

~W c_ f(V/~) . Thus

the

set

(3)

we have

+

c(hi,qhi)i E Yi Since

while

H E F , it

C(K'qK)i ~ Yi ;

follows

that

hence

q ~ f(V/F)

.

(c q)i ~ V.I " This

completes

the

proof.

Our final version of 1.7.2 concerns regularity.

Lemma 1.7.6. Rep(e) . in

~i

Also suppose

and

assume that

a E PiEIAi

~a i = ~f(a/F) .

Proof.

show that

Assume the hypotheses of Lemma 1.7.2.

Then

and for each f(a/F)

q E f(a/~) .

q E f(V/~) , and

Let

and

is regular

F = I U Af(a/F) , and

FI P = FI q .

Let

+ H = [i E I : (c P)i E a i

ai

f =

is regular.

We assume all the hypotheses. P E f(a/~) ,

i E I ,

Let

+ (c q)i E Vi} -

We want to

.

1.7.7

Ultraproducts

Thus F I

H E F .

(c+P)i

implies that

+ e , for all

By the definition of

+

= F I (c q)i

Thus for any

+ (~ q)i E a i

Remarks 1.7.7.

Since

95

i E I

we have

i 6 H , the regularity of

H E F , it follows that

ai

q E f(a/F)

.

The above lemmas are algebraic forms of the ~o~

lemma for ultraproducts.

The exact relationships between them and

~o~'s lemmm will be discussed in a later article. Andr~ka and N~meti have shown that various hypotheses in 1.7.2-1.7.6 are essential, and that these lemmas do not generalize to arbitrary reduced products.

Now we shall use the above lemmas to prove various closure properties of our classes of set algebras.

Theorem 1.7.8.

Proof. UpK

UpK = IK

for

K E [Gws

By Lemmas 1.7.2, 1.7.4 (i), and 1.7.5, each member of

is a subdirect product of memebers of

IK = SPK ; if

,Gs ] .

~ = I

K .

If

~ ~ I , then

, then by 1.7.3 and 1,7.4 we have

UpK = IK ; the

proof is complete.

Theorem 1.7.9.

Proof.

UpCs

directed by

{I(Ws

UL E K

~ , for

N Lf ) :~

Proof.

for

~ < ~ 9

By Lemmas 1.7.3 and 1.7.4 (ii) .

Theorem 1.7.10. K

= ICs

an

whenever

K E [l@ws

ordinal]

L

is a non-empty subset of

,IGs ] U [ I C s

: ~ < ~] U

.

This is immediate from 1.7.8 and 1.7.9, since

for all choices of

K

except the last.

Let

K = Ws

n Lf

SK = K , , and let

96

Ultraproducts

L

be as indicated.

UL

Then

1.7.11

UL E Gws

i s simple by 1 . 5 . 2 ( i i ) ,

by what was already proved, while

2 . 4 . 4 3 , and 0 . 3 . 5 1 .

Hence

/JL E IWs

by

1.6.4.

The following generalization of a part of Theorem 1.7.10 is due to Andr~ka and N~meti

[ANI]:

Theorem 1.7.11o , then

let

~

be an isomorphism onto a regular

such that

P~6.L~/~

~

~

= [~ E L :~ G ~] .

E F

onto

b E P~EL B 9

for each

P~EL ~ / ~

Let

c

Cs

be an

with base

9 E L .

such that

Let

~ ~ w 9 Cs

F

~

For each

with base

be an ultrafilter on

There is an isomorphism

g(b/F)=

( ~ b ~ :~ E L)/~

(F,U,~) - choice function, and let

By Lenmms 1.7.2 and 1.7.4 (i), onto a

is a set directed by

By Theorem 1.7.10 we may assume that

U~ ; further, let L

0 ~ L ~ ICs reg n Lf

UL E ICs reg n Lf

Proof. E L

If

f

is a homomorphism from

P~EL U~/F .

Now for each

g

of

for all f = Rep(c) .

P~EL ~ / ~

b E ~EL B

let

~) h'b = (b : b E B, ~ E L) U (0

and let P~6L~/F

hb = hlb/~ .

Then

h

f o g oh

is simple, by 2.3.16 (ii).

2.1.13,

~ ELf

b E U~EL B 9

Now let

is an isomorphism, and

First note that each member of DL

To show that

Then for each

E L),

is an isomorphism of

(see the proof of 0.3.71).

We claim that

:b ~ B , ~

Thus ~

9 E L ,

L

UL

into

~ = (f o g oh)* UL ~ E Cs reg n Lf

is simple, by 1.5.2 (i); hence

f o g oh

is an isomorphism.

is regular we apply 1.7.6. ~(hJb)~

is regular in

By Let

~

, and

1.7.12

Ultraproducts

~(h'b)~

since

07

= ~ ( h ' b ) ~ G ~ b G ~(fghb)

f ~ g oh

is an isomorphism.

Therefore by 1.7.6,

fghb

is

regular, as desired.

The following

lenmm is due to A n d r ~ k a and N~meti.

Lemma 1.7.12. V , and let

F

Let

~

be a

be an u l t r a f i l t e r

(F,
.

Define

Crs

with base

on a set and let

6 E A(IA/F)

We assume also that for every all

K < ~ ,

= g*~ . (i) (ii) from

be an

e E U(Iu/~) by

,

there is a

c(K,E(rK)) ~
g ~*~

is an i s o m o r p h i s m from is sub-isomorphic

~ = r6~

o g

.

such that, for

Z E F

Finally,

let

Clearly

onto

to

~

g = f o6

and

;

(recall the d e f i n i t i o n of

from 1.3.15);

(recall the definition of

g

is an isomorphism from (ii) will follow.

is a h o m o m o r p h i s m ~

onto

~ ~

To establish

show

(I)

c

Then

Proof.

(i) and

Let

~U>

r E V

1.3.5, and s u b i s o m o r p h i s m (iii)

and unit element

f = Rep(F,,~,

and

6 = < = <
I .

U

ga N E~V = [~ o s : s E a}

.

from

r6

~

from 1.6.1).

onto

~

, so if we establish (iii), let

a E A

.

By 1.3.1,

(iii),

; we want

then to

98

Ultraproducts

First

suppose

q s ga n ~ V

that

q = ~ o s .

that

c(K,~(sK))

.

Since

q E ~V

By the h y p o t h e s i s ~

<sK : i E Z>

1.7.13

, there

is an

of the lemma choose

for all

K < ~

.

Let

s E V Z E F

such such

H =

+ [i E I : (c q)i E a] i E H n Z .


q = ~ o s Second 9

Since

<s~ : ~ < => = < = ( ~ , ~ ( s ~ ) )

=

q E~V

H E F .

H n Z E F , we can choose

Theu

s =

Thus

; thus

: ~ < c~> =

implies

suppose

that

(c + q ) i

q

q = E o s

i : ~ < c~> s a

.

is in the right with

A g a i n by the h y p o t h e s i s

s E a

.

side of

Since

of the lemma

let

(I).

a ~ V Z E F

, we have be such

that

+ c(K,EsK) ~ i 6 Z

<sK :i E Z>

, so

q E ga ~ V

We n o w use to A n d r ~ k a

Proof.

Let

!~I U 9

I

[ CK],

9/

Monk

Ws

take

6

and

~

(c q)i = s 6 a

for all

Ws

= ICs reg

the fact that

w i t h unit

of

Let

F

K - regular

as is e a s i l y such that

seen,

function

element choose

he a

Ws

, also due

~ ICs

,

V = ~ U (p) I

ultrafilter there

and set

satisfying

.

Let

in a special

way;

I I l - reBular u l t r a f i l t e r see Chang,

Keisler

is a function

[i E I : K 6 hi}

be as in 1.7.12,

(F, , ~ ) - c h o i c e

Ws

[HM]

I = ~).

h 6 I[F _c ~ : IF 1 < ~} N o w let

that

(in a later proof we shall

Then,

Then

_c iCs reg

be a

(for the n o t i o n p. 201).

.

to e s t a b l i s h

It generalizes

1.7.13.

for n o w we could over

this lemma

in Henkin,

Theorem

K < ~

.

and N~meti.

established

IIl ~

for all

E F

X = IU/F

for all .

the f o l l o w i n g

Let

K < ~ 9 c

be an

condition:

99

I. 7.14

(I)

For all

K < ~ ,

i E I , and all

o(K,y) i = pK ; if

K E hi

Let

We shall show that

f = Rep(c)

By 1.7.2, fSV = ~ X

.

f o8

and

~

is a

show the other inclusion. for all

If

K ~ hi

hi

is finite,

y = ~u

with

f ~8

is a h o m o m o r p h i s m onto a

, so that

+ (c q)i E V

Y E X , if

Cs Let

i E I .

, then by

(i)

it follows

To show that

f o8

Since q E ~X .

So, let

we have

Crs

K E hi

rK = pK

and

hypotheses

(~N

trivially, we

Note

that

(c+q)i E ~U

c(K,r

c(K,r

c(K,ErK)i = rK

It remains to show that

h

Let

we have

= rK = rK

by

(I) .

by either clause of

~

r E V .

We have Then there

F) I P 9

Let

Z =

Z E F .

Let

K <

(as desired).

f ~6

If

If

K ~ F (I) .

K E F , then

Thus the

is an isomorphism.

is regular; but since

f o6

is an

this is immediate from 1.7.6 and 1.1.16.

9T h e ~

1.7.14.

Proof.

Using

__ reg I G s = IGs reg~ = l ~ s = IGws

I.I.I0,

I.I.II,

1.1.18,

~ SPCs reg

1.1.19, 1.6.4, 1.6.6, and

1.7.13 we have

IGws

Since

, as desired.

F) I r = ( ~

of 1.7.12 hold, and hence

isomorphism,

Now we show that

(c+q)iK = c(K,qK) i = pK .

By the choice of

and so

.

is an i s o m o r p h i s m we apply 1.7.12.

i E Z ; we show that

then

~

i E I .

such that

and

c(K,y) i = u .

It suffices to show that

is a finite

.

then

fSV ~ ~X

of 1.7.12 left to check.

[i E I :F ~ hi}

then

is the desired isomorphism.

only one hypothesis F ~ ~

u E U

+ ~u(p) (c q)i E

that

K ~ hi

^p~ reg ~eg c SPWs~ ~ ~ ~s~ _c IGs _c IK _c IGws~ c SPCs~ eg ,

.

100

Ultraproducts

where

K = Gs

The long

Gw8 r e g .

following

time

using

to A n d r ~ k a

theorem

follows.

IGs

HGws

representation

, was The

theory.

Proof.

The

, and

9//L E I G s

HGws

case

Let

F

all

z E L

let

L

is trivial,

be an ideal

of

so a s s u m e

9/ .

I.

~ < w

.

For each and

be an u l t r a f i l t e r .

Now

a E A

~gkz~/

over

define

h

z E A

proof

is due

~ > 2

.

to show

Let

that

E Gs

.

such

that

L

mapping

A

(a 9 kz : a E A>

we

infer

9

.

Note

that

= <~gkz~/ : z E L>

Iv E L : v > z] E F PzELBz/~

by

.

for

setting,

a 9 -c(~)v

it s u f f i c e s = 0

; since

On the o t h e r

if

hand,

a . kz # 0 ; thus

by

h E Hom(9/,PzEL~z/~)

this

for all

.

z E L

E Hom(9/,PzEL~z )

[v E L : a 9 k v = O]

Thus

Let

into

E Ho(9/,~ z) for e a c h

that

h 9/ ~ 9//L ; to show , then

kz = -c(~)z

,

< : a E A>

hence

We w a n t

let

ha = / ~

a E L

direct

for a

~ IGs

~ < 2

is z e r o - d i m e n s i o n a l

O.3.61

present

to the a u t h o r s

.

Case

Now

known

and N ~ m e t i .

1.7.15.

for a n y

The

result,

Theorem

9/ E Gws

kz

or

I. 7.15

9

to show v > a

and

2.3.26,

0.3.6(ii).

Iv E L : v > a ] E

a E A N L

by

Hence

N o w we c l a i m that , so F

z E L

so

h-l~o

by

that

= L

.

If

Iv 6 L : v > a] =

, it f o l l o w s then

that

a ~ c(~)z

ha = 0 .

and

ha # 0 , as d e s i r e d .

h 9/ ~--9//L , so

9//L E S U p G s

By

1.7.8

,

9//L E IGsc~

1.7.15

Ultraproducts

Case 2. a E A " L ha # 0

~ ~ ~

and

and

.

Using

1.6.3,

find a h o m o m o r p h i s m

h L = [0]

101

it is enough

h

of

let

F

& ~ F} 6 F

be an u l t r a f i l t e r for all

onto a

Cs

such that

Let

I = L• [ r ~ - ~ : l r and

~

to take any


on

I

E I .

such

Let

<|

I

that

9/

[
have base

E I :v > z,

U

and unit ele-

ment

(Pj)

V = U[ (Pi) ~ (Pj) y. n Y. = 0 i 3

where Since

a ~ L , there

-c(F)z that

for all r.l E

: j E J]

for distinct

is a function


(Pji) Y.. jl

Y. J

E I .

for all

Then

i,j E J 9

r E IV there

i E I .

,

such

Let

that

r(z,F)

is a function

N e x t we

X = IU/~

.

E a 9

j E Ij

such

let

Q = [ k / ~ : k E PiEIYji ] , s = <
Let

c

be an

conditions

(F,
< ~)

function

such that

the following

hold.

(I)

If

K < ~

and

y E Q

, then

(2)

If

K < ~

, then

c(K,sK/~)

(3)

If

K < ~

, y E X

, (z,F)

c(K,y) = sK

E PiEIYji

.

E I , and

K ~ l" , then c(K,y)z F =

r(z,r)~ Let

f = R e p ( F , < U : i E I),~,
1.7.12

.

Then we c l a i m

that

f o6

.

Also

is the desired

let

8

be as in

homomorphism.

First

102

Ultraproducts

we note that by the second part of Next we show that v > z]

; thus

+ (c q)i ~ z where

(f ~

Z E F .

(thus

v ~- z .

L = [0] Let

1.7.2 and by .

By

(3)

= ffQ

(4)

q E ~X

q E ~X

other hand, By

let

and

i =
it follows

that

(c+q)i ~ z . (f o 6) ~

is a

i = (z,r)

Cs

; w e show that

for all

iff

(I) ,

(c+ q)i E ~Y.ji

i E I

and hence

for each

and all

(3)

(4) . for all

q E f6V

.

i E I , On the

+ Z = [i E I : (c q)i E V]

Then the set

K < ~

(c+q)i E ~Yji

we h a v e by

A n easy a r g u m e n t yields

(c+ q)i E ~Y.ji

i E Z

i E Z .

is

i.e., Thus

q E ~Q

9

This

the proof.

T h e o r e m 1.7.16. HSPWs

r i ~ c(F)v

i E I , then with

q E f6V .

(4) ,

c(K,qK) i E Yji completes

i E Z , that

for all

+ (c q)i E V

Now suppose q 6 ~Q 9 By + so by (4) (c q)i E V for all

F .

[(v,F> E I :

i E Z , say

i E I ,

and any

+ (~ ~ r)1 (c q)i = (~ N F)Ir i

in

Z =

f6a # 0 .

.

For any

For, if

Let

Since

v > z ,

Set

We show, for any

c(K,qK)i = r.K l

we have

and, since

(2) , we have

z E L .

q ~ f6z , as desired).

It remains only to show that f6V

Let

q E ~X .

+ (~ "~ F)I (c q)i = (~ ~ F)Ir i

thus + (c q)i ~ c(r)v

I. 7.16

= HSPCs

Proof.

For

~ > I

we have

IGs

= HSPGs~ = H S P G w s

=

= HSPCs reg

HSPGws

= HGws

= IGs

c IGws

using

parts of the theorem are easily established by using

The following result is due to Monk.

1.7.15; 1.7.13.

the other

,

1.7.17

Ultraproducts

Theorem

1.7.17.

A n y direct

Proof.

We may assume

103

factor

of a

Cs

is isomorphic

to

Cs

a

pressed

Gws

in

1.2.6.

that

interesting

(I)

= [~ :~ is isomorphic

compressed

The

Gws

theorem

zero-dimensional element Say

result

element

the unit element

of

the notion

from

~

on

is

V

~

such

that

(2)

for all

Let

f = R e p ( F , ( U : ~ < ~),~,
.

be an

and all

are met, and hence

let

~

the following

Gws~}

2.4.8,

,

the unit

be a c o m p r e s s e d

and

is 6

c(K,gu

: ~ < ~),c)

f ~

any

since any

U

.

be as in

F

be a

1.7.12.

function

=

Then

Gws

Let

(F,,~) - choice

u E U

inde-

; in fact,

is clearly

and its base Let

com-

and N~meti.

(I), using

Now

of

Cs

Gws

X = ~U/F

K < ~

c

to a

Gws

Let

1.7.12

Let

by A n d r ~ k a

of a compressed

~

ultrafilter .

We use

to a c o m p r e s s e d

immediately

of a compressed

I ~ I - regular

noticed

is subisomorphic

follows

9

In fact, we first prove

pendently

ICs

~ e w

.

the h y p o t h e s e s

is an isomorphism.

N o w by

of

1.7.4(i),

* (f ~ see

~ E Gws that

Qj = Qk

if we examine

the proof

for all

j,k

in its

and hence

(f ~

that proof

that the base

If

we are finished,

W = 0

~

Actually

E PiEiJi

is a c o m p r e s s e d

self is the unit element

of

(f ~

*~

so assume

of some

Gws

Gws is that

too. X

.

of

1.7.4(i)

notatio~with Also Let

W @ 0 .

recall

W = fiX ~ Note

N o w we claim

that

we

I = ff , from f6V W

. it-

104

Ultraproducts

(3)

9/ > ~

In fact, and

applying

Gws

U

proof

of

Since

~

By

1.7.13

such that

from

9/

and b y

we

9/ we ~

find

~ > ~ ~

1.6.2

, as desired

to a

(3)

for some Cs

that

we get a in

.

w i t h unit e l e m e n t

9/ > ~

is isomorphic

~

~

looking

the base of

Gws

; let

;

Ws

~ g

V

wi~ at

~

the

is

X .

w i t h unit element be a h o m o m o r p h i s m

. 9/

~ X (f ~

is isomorphic

9.I

is

h = < g a U f6a : a E A}

It is e a s i l y

that

W

Gws

see that we may assume

0.3.6(ii),

function

to the full

is zero-dimensional,

onto

By

to

1.7.13,

W ~ ~X

w i t h unit element

2.3.26(ii)

then restricting

base

W

for some

1.7.18

checked

that

to a subalgebra

isomorphic

to a

Cs

is an i s o m o r p h i s m

ha n'~v

= ~a

for

of

9/i

; in

from

all

~ x (f ~

~1

fact,

into

a E A , so

9/ ,

the ~/'

(1)

holds.

We and

shall e s t a b l i s h

relationships

between

after

discussing

about

the results

Remarks

again

the obvious ways; (I) 2 ~ IUI < w such

that

If

~/~

does n o t extend and choose

and

~ ICs to

9/

is a

is a set This was

IUI > ~

an u l t r a f i l t e r

several

, by F

over

in this

above

specific

in

1.7.28-1.7.30,

some

cannot

U

by M o n k

In fact, that

in

to this effect.

such that

and an ultrafilter

such

remarks

be improved

arguments

w i t h base

1.7.22.

properties

section.

first noticed

I

closure

First we make

Cs I

about

of set algebras

of the results

we now make

, then there

of base.

established

Many

~ ~ ~

results

our classes

change

already

1.7.18.

a few more

let

I IA/~I

F

over

[M2] I = 22~Z = 2221~

I It

1.7.19

Ultraproducts

(see, e.g., Chang, Keisler

[C~

IUl , and hence so does

~/F

has base of cardinality

IUI

Thus

I~/~ ~ I C s (2)

~ ~ w

ultrafilter

~

Now

But any

~

Cs

has characteristic

of characteristic

, and hence has at most

, any n o n - d i s c r e t e

F

x =
|

p. 202) 9

221~l

This same remark and proof apply to

For any

prinicpal

.

I05

on

w

Then

we h a v e

~ E Ws

and

elements. !

Ws

s .

, and any non-

w N / ~ ~ IWs

0 < x/~ < I

IUI

For,

let

~(x/F) = 0 .

By

1.6.13,

1Ws

Now we again discuss change of base ultraproducts.

(see section 1.3), using

First w e prove a sharper form of part of 1.7.13,

due

to A n d r ~ k a and Nemeti.

T h e o r e m 1.7.19. base

U .

where

X

x E A

.

Let

let

Then

F

notation

~ ~ 2 .

be a cardinal

Let

~

is sub-isomorphic

Let

!I I -reguiar

in the proof of

be a

such that

to a

Cs reg

9.1 have unit element

be a

~

Ws

with infinite

<,~-. IAI 9 IUI < ~ / ~X~

such that

is the least infinite cardinal

Proof. and

y

Assume

on

for all

with base of power

~ U (p)

ultrafilter

IAxl < X

,

Let I

.

~ .

I = max(~,y)

Introducing

the

1.7.13, we see from that proof that

f o6

N*

is an i s o m o r p h i s m to

~

[CK]

.

of

Also note that

we have

a subset

W

isomorphic W'

onto ~

a

csreg

has base

IXI = IIuI > y . of

X

is e x t - i s o m o r p h i c

from

~.i

such that to a

to

~ .

onto

W

, and

X .

E ~I

is

sub-isomorphic

By P r o p o s i t i o n

4.3.7 of

Hence we may apply 1.3.18(iv) ~ U _c W ,

Cs r e g ~

Choose a set such that

~

with

~ c a'

IWI = y , and such that

base

W' ~ U

to get

W .

Thus

a n d a one-one Let

E ~I

is

sub-

function

~ = (a'-l)~*~

~'

(cf. 1.3.5).

106

Ultraproducts

Thus

~ E Cs reg

by

1.3.1

is sub-isomorphic

to

and

1.7.20

1.3.7, and it is easily checked that

~ .

Using this theorem we can prove one of the basic results about set algebras,

due to Henkin and Monk.

tion used in

1.5.6.

Definition 1.7.20. K

the class of all

~ 6 K

Theorem 1.7.21.

Proof. Vi =

where for all

Let

a E A ~" [0] Via

h a E Hom(9/,~ a)

For

and

K s [Cs ,Gs ,Gws ,Ws ]

~ m m

Gws

~a

For each

a E A

z a 6 Jaha a

to a

for each

for each

set

a E A

let

~a

ga = r2"x o Ja Oha

2 K = Ua6AWa

for distinct

Cs

Ca

with base

a E A .

a 6 A ~ [0]

X a = ~(2K) (za) , and let every

~a

Thus

V i N Vj = 0

be the full

Ws

w~th

Thus Let

Choose o

,

a,a' s A .

IWal = 2 K for each By

1.7.19

W a , for each z E A(s(2K))

For each

a E A

Ors

a 6 A -- [0]

with unit element

ga 6 Hom(9/,~ a)

for all

, ; let

such that

X 0 = s(2 K) ~ ~ IX a : a 6 A ~ [0]]

be the full

let .

For Xa

and

a s A , and

a

ga a ~ 0

for

a # 0 .

Hence

Pa6A~a E ICss , as desired.

,

U IAI U Isl

be such that

Wa n Wa , = 0

Ja E IS(~a,~ a)

let

UiElVi

i E I , and

ha = r6~. (recall 1.6.1). la a E A , and h a a ~ 0 if a ~ 0

for all

as isomorphic

~ I Cs

, and set

<Wa :a E A>

a C A , and

=Gws

, say with unit element

K = UiEllUil Now let

we have

U i is infinite for all

.

we denote by

having all subbases infinite.

9/ be a

(Pi) Ui

unit element

For

First we formally give a defini-

~/ ~-- I c Pa6A~a

.

By

1.6.2

we have

1.7.22

Ultraproducts

Remarks 1.7.22. closed class for

By

1.7.16

~ m w .

and

showing that any

CA's

.

Gws

1.7.21,

I Cs

Thus for many purposes,

with infinite bases are the simplest represent abstract

107

CA's

is an algebraically

cylindric

set algebras

with which to isomorphically

Andr~ka and N~meti have improved

is sub-isomorphic

to a

1.7.21 by

Cs

Before turning to results concerning change of base by increasing the cardinality of the base, algebraic analogs of the upward L~wenheimSkolem-Tarski The example,

theorem, we give an example supplementing I

K < m , then there is a then the base of

~

k < K

base

K o

9/ ~

E Cs

~

Now

K

~

I < ~ < ~

such that if

(whether or not

9/ ~

and 6 Cs

K < ~ ); because of

in most of our theorems that the base is infinite. a k = [
U

Since

shows that if

with base

of

~

Let

9/ be the full C s

are the distinct atoms

f E Is(9/,~) o

, so the base

X < K .

9/

a0,...,aK_ 1

, say

IU] > K .

C(C ~

let

Thus

Cs

has power

this example we assume For each

l

due to Andreka and Nemeti,

those of section 1.3.

Then

fa0,...,faK_ I

has at least

k
< ~

K

(c0a k ~ ak) [] c(o~l)a k = 0 , so

Now suppose

are distinct atoms

elements.

it follows that

.

with

Now suppose

I fa%l > I

for some

(c0fa k," fa k) n

,~l)fa k = 0 , contradiction.

The following general lemma and theorem about increasing a base !

I

are due to Andreka and Nemeti;

their parts dealing with

Ws

and

Cs reg

are due to Henkin and Monk.

Le-r~ 1.7.23. V .

Let

F

there is an

be a

Let

~

be a

Gws

with base

I~I- regular ultrafilter

U

on some set

(F,,~)- choice function

c

and unit e ~ m e n t I .

Then

such that, letting

108

Ultraproducts

f = Rep(F,,~,,o) in

1.7.12,

6

and letting

and

~

be as

we have:

(i)

f o6

(ii)

~ 9/

(iii)

1.7.23

is an isomorphism is sub-isomorphic

from

9/

(f ~ 6) 9/ ;

onto

to

(f o 6)

is

IU/F

;

~ = r6~ o f o 6 ; aV

(iv)

~ V = ~(E*U)

(v)

the base of

n f6V ; (f ~ 6) 9/

;

w (vi)

9/ E K

implies

K 6 [Ws

Proof.

(f o 6) 9/ 6 K

,Cs ,Gs ,Gws

Assume

for distinct

X = IU/F

o

R = {<j,k>

relation

on

Let J .

reg ,Gws~ ] .

the hypotheses.

~Y!PJ)3 n ~y~pk)

Let

K

for all

V = ~Hj E J - -~Y(PJ) j

Say

j,k E J .

Let

E 2j :yj = yk ] .

be a subset of

in co~Inon with each equivalence

J

Y = Thus

R

class under

R .

Let

in common with each equivalence

~ ( < K :is

<j : i 6 I> 6 L

functions

w s X(Iu)

and

for each

v E X(Ij)

.

and

one element

L ~ IK

have

class under

j E K .

Let

O =

is an equivalence

having exactly

exactly one element , with

' where

y E X .

Now we define Choose

k

= Y

k 6 IK some

such that u 6 U .

y N PiEIYki @ 0 , and

Let

y N Pi61Y(vy)i

vy

if

y = Eu .

Now

wy s y

for all

(2)

weu =

w

and

y 6 X ; for all

of

if

L N k/F

wy s y ~ Pi61Y(vy)i

properties:

(I)

constant

be the unique element

~ 0 , so we can pick

wy =

k

u E U ;

v

y = Eu .

Thus

with

have the following

for

1.7.23

Ultraproducts

(3)

for all

(4)

if

Since

y s X

k,q s Rgv

F

is

and

i E I

and

we have

(wy)i 6 Y ( v y ) i

[i E I :Yki = Yqi ] s F , then

1 5 1 - regular,

[i E I : K 6 hi] 6 F

109

choose

for all

N o w let

K < 5 ,

P(vy)i K

k = q .

h E I[F ~ 5 : Irl < w]

K < 5 .

choice function such that for all

;

if

c be an

(F,,~) -

y 6 X , and

K ~ hi

and

such that

i 6 1 ,

y ~ e U ,

c(~ 'Y)i L(wy)i

Note that

c(K,y) 6 PiEiY(vy)i

otherwise.

for all

be as in the statement of the lemma. hold. For

(iv), use

be the base of obvious

that

in fact say

Z = X .

q 6 5X , and f o r any (2)

By 1.7.12,

(2) and the definition of c.

(f o 6) ~

u E Yj

K < 5

; we are to show that Suppose

with

y E ~ U ; say

j E J 9

i E I

Now let

c .

y E X .

(i),

Let

f,6,~

(il) and (iii)

To prove

(v), let Z

Z = X , and it is y = s

with

u E U ;

q = <EpjK : K < 5>~

9

Thus

(c+q)i = (pj)~ E 5YIPJ) ~ V using

we have

and the d e f i n i t i o n of

and

It follows

that

q E f6V , and hence

. y 6 Z . i61

N o w let

y E X ~ e U .

Let

q =

9

Then for any

,

(c+q)i =

= ( 5 - hi)IP(vy)i U <(wy)i : K E hi> E ~Y(vy)i (p(vy)i) c_ V . Hence a g a i n

q E fSV

and

y E Z .

N o w we turn to the parts of K = Cs

5

(f o 6)*~

are

taken care of by 1.7.4.

b y 1.7.6,

since

f ~6

So

(v)

(vi) . If

~

holds. The cases

K = Gws 5

is regular,

is an isomorphism.

and

then so is

N e x t suppose

110

Ultraproducts

K = Gs

Thus for all

For each

r E Rgv

(5)

if

For,

suppose

Then

j,k E J

we set

r,r j E Rgv

Qr

and

1.7.23

we have PicIYri/~U

r @ r I , then

y 6 PiEiYri

,

(6)

fSV ~ ~r ER gv Qr

"

For,

let

Thus the set

We claim that

q E ~Qvqo

(c+ q)i E ~y ji " and also tion of

.

~

c(K,qK) i E Y(vqK)i

[i E I :Yi = zi} E F .

r = rl

i E M

i E M

Yji = Y(vqK)i

, and

+ M = [i 6 I : (c q)i E V}

and

(c+q)i K = c(K,qK) i E Y(vqK)i c , so

Yj ~ Y k = 0 .

Qr N Qr I = 0 .

(4)

For each

Now for any

or

Now

z E Pi61Yr~i

[i E I :Yri = Yrli ] E F , so by

q E fSV

Yj = Yk

"

choose

K < ~

"

ji E J

we have

Thus

i E M

and

qK E QvqO

F .

so that

(c+q)i K 6 Yji

by the note following

So for any

= Yji = Y(vqO)i

is in

the defini-

K < ~

for any

we have K < ~

as desired.

(7)

If

r E Rgv

For, choose c

,

K < ~ , and

z E Y ~ PiEiYri

we also have

c(K,y)

{i E I : z i = c(K,Y)i] we infer from

(8)

For all

In fact, c(K,qK)

let

(4)

q E ~Qr

E PiEiYri

-

' then

is in

F o

.

By

Hence 9

c_

~Q

(7)

r

Thus

E PiEiYri

9

c_ f6V

the set

M =

follows.

.

for all

+ = i 6 I , (c q)i

Hence

of

M ~ {i E I :Yri = Y(vy)i } '

(7)

we have,

for all V .

9

Since

vy = r , so

we have

c(K,y)

By the remark after the definition

6 y ~ PiEiY(vy)i

that

r E Rgv

9

Y E Qr

q E f6 V

.

K < ~ ,

,

1.7.24

Ultraproducts

By (5),

(6),

(8)

In the case #

we have

K = Ws

(f o6) ~ s Gs

we redefine

K < ~

let

K < ~ .

We may assume c(K,~u)

Choose

that

IUI > 1 .

=
k

6 y

and let

Y

E U - [pK]

9

.

Thus

with

the additional

(9)

for all

To establish

and

f6V = ~X (e~

is an

suppose

for all

c(K,qK)i

Thus

function,

so

pK ~ Rg c(K,y)

(i) - (iv)

it suffices Let

to show

we have

Hence

~ pK

by

F

Then F

is

= c(K,Ep~)i

=

q E f6V 9

q 6 ~X ~ ~X (E~ ; thus

that

Then

(c+q)i ~ = c(K,qK)i

.

are true.

i E I .

F = [K < ~ :qK ~ cpK]

i 6 I , and consequently This completes

we have

q 6 ~X (eop)

(c+q)i E ~U (p) = V .

we have

.

= Zy Ik Y U , where

clearly hold,

Let

F = [K < ~ :qK ~ epK]

i E I

= [i E I :k i ~ pK] Y

(F,,~) - choice

suppose

K E a ~ F

Conversely,

and

(f ~ 6) ~ 6 Ws

(c+q)i E ~U , obviously.

Thus

For each

Y E X ~ [~pK]

1.7.12

First

and for

Now suppose y E X ~ E U

c(K,y)

and all

of

Let

, say

property

K < ~

(v)

Let

c

Now the hypotheses

pK o

~ EWs

and

Y

since y ~ s

finite,

Let

u E U

Z

Y

u

c .

%

V = ~U tp) .

gy E F

11~

; we show that

is infinite.

(9) .

Therefore

q ~ f6V .

For any

+

K E F

(c q)i ~

~u(p)

and

= v

q ~ f6V .

the proof of

1.7.23.

This lemma imm~ediately gives

Theorem 1.7.24. let

~

be any

Let

cardinal.

base of cardinality

~ Then

be a ~

~ K , such that

Gws~

with an infinite

is sub-isomorphic

to a

1~ = 1~ n ~W

for some

base, and Gws

~

W

.

with

Ultraproducts

112

. reg reg reg. 9/ 6 K E {Ws ,Cs ,Gs ,Gws ,Gws ,Cs ,Gs

Moreover, if also

1.7.25

then

9 E K .

Using 1.3.18 we easily obtain the following more specific result.

Theorem 1.7,25. and suppose that Gws

~

some

Let

~

be a

IAI U IUI ~ K .

with base of cardinality W o

Gws

Then

with an infinite base 9/

is sub-isomorphic to a

K , such that

191 = 1~ ~ ~W

for

Moreover:

(i)

if

9/ E K E fWs ,Gws ,Gs ,Gwsreg,Gsreg]

(li)

if

9/ E C s

(resp.

~ECs

U ,

Proof.

(resp.

~ E Cs~ eg)

and

then

~ 6 K 9

K = K I=I , then

9 E Cs reg) 9

The parts concerning

Ws , Gws , Gws reg

are immediate from 1.3.16, 1.3.18 and 1.7.24.

Cs

and

Cs reg

For the other parts

reg dealing with

Gs

and

Gs

we use a direct construction, not

involving ultraproducts (which actually works for some other classes). Suppose

9/

is a

Gws

UjEj~Yj (pj) , where Let



IUI

for all

onto

ZB

UI Id .

V [~ c

f~

= 0 .

Let

<~ UjEj~(f$Yj) (f~~

6 ~Z B : fB 1 o Y 6 x].

for each

V =

j,k 6 J

9

Z0 = U

be the full

For each

c E C , i.e.,

~

of

and

U

f0 =

we have

x E A

It is easily checked that

onto a subalgebra

IZ~I =

be a one-to-one function mapping

,<~,k~ E K x J

n ~(f ~yk )(f7~

~

for distinct

~ < K ; further, we assume that

Then for distinct

unit element

and unit element

be a system of pairwise disjoint sets, with

for each

morphism of

U

~Y(PJ) n ~y(pk) = 0 3

~ < K , and let

~(f Syj) (fB~

~
, say with base

let g

~ , and in fact

9/ is sub-isomorphic to

Gws

with

gx =

is an isog

-I

~ .

c = Moreover,

1.7.26

the base of

~

is of power

Finally, suppose and

113

Ultraproducts

K .

~ E Gws reg

In case

Suppose

(~gx U l)ly = (~gx U l)Jz .

while

z 6 ~Z

Thus since

and

A x = ~gx , because

by regularity of

For

x

and

Gws's

Since

g

hence

x E A y 6 =Z ~

Say

f-I o z E V . Y

~ E Gs~ , clearly

~ E Gs

y E gx

z 6 gV

f~ I o y E X ,

and

yO = zO

we have

8 = y .

is an isomorphism, we get

f;1

oz E x

z E gx .

in general it seems to be more interesting to increase

the size of various subbases rather

t~an to merely increase the size of the

base; that is the purpose of our next theorem.

Theorem 1.7.269

j,k E J 9 Let J

such that

j E J

Let

~

be a

V = UjEj~Yj (PJ) ' where

element

and

Yj = W.j

lwjl =

~Y(PJ) N ~v(Pk) = 0 j -k

and

~

is sub-isomorphic

, with

for all

to a

~W j(pj) n ~w(Pk) = O -k j E J j E J

for which such that

Gws

and unit

for distinct

function with domain Kj m IYjl ~ m

~

whenever

with unit element

for distinct

Yj 1

Kj = 1

Kj > IYjl

, and

9

j,k E J , where W.j ~ Yj

with

Furthermore,

.

By 1.6.2 we have

9/ ~ I - c Pjs

with unit element is given by

that

U

Kj ~ IYjl .

Proof.

h

with base

be a cardinal-number-valued

for all

v=%nl

Gws

Kj ~ (IAI n 2 I~IUIYjl) U ~

Then U j E j ~'(PJ) w j

K

IYjl ~ ~

9

~j

, where

~j

is a

Ws

~Y(PJ). for each j E J ; in fact, the isomorphism JI (ha)j = a n ~Y[.PJ)j for all a E A and j 6 J 9 Note

and

IBjl IA! n 2!IulYjl

114

Ultraproducts

for each

j E J

for which

isomorphic to a

Ws

yj = Wj

and hence

We may assume that

IYjl ~ Kj .

Hence by

with unit element

Wj n W k = Yj n Y k For each

f. E Is(~j,~j) . J

j E J

H

g

0~(pj)

~JEJ"j

~

and

~j = ~j

for all

j E J 9

~W~ pj) n ~W~ pk) = 0

aEA

;

onto a

(as is easily checked)

is sub-

fj =
let

Finally, let for each

is an isomorphism from

~j

such that

1Wj I = Kj

and hence

ga = ~jEjfjl(ha)j

then

1.7.25,

~W~ pj)

Kj = 1Yj I ' while

if

J,k E J 9

for distinct thus

~. J

1.7.27

g

Gws -I

~

with unit element

d = V n d

for all

d E D ,

as desired.

We now return changing the function

to a question discussed in 1.3.20-1.3.22: p

in the unit element

~U (p)

of a

Ws

The

following theorem of Andr~ka and N~meti strengthens results of Henkin and Monk.

Theorem 1.7o27. element

Suppose

~U (p) , and let

with unit element

~Y(q)

IN,- Rgql = IUI ~ IAI

~ ~ w 9

q E ~U . with

F

then one may take

he an ultrafilter on

all

A E I ~

Let

X = IU/F .

all

K < ~ 9

Let

E

K E~NF

,

l I

be a

c

with unit

IUl < ~

Ws

or

Y = U .

l>l.

k E ~U

be as in 1.7.12 .

Ws

is homomorphic to a

Let

such that

Choose

(F,
~

~

U c_ y , where if

Proof. We may assume that and let

Then

Let

[~ E I : A = F] E F so that

kK ~ pK

for

for

Then there is an

such that for all

y E X ,

FEI,

1.7.28

Ultraproducts

~pK

if

c(K'Y)F = ~ikK

Now let

y = eqK ,

otherwise

.

f = Rep(F,,~,
a homomorphism of 1.7.12 Now if

115

from

19//~ onto some

is an isomorphism of

h E ~X

and

9/

F E I , then

Crs

~

into

. .

Thus by 1.7.2,

h 6 ~X (~~

Choose onto t

Y ~ U X

from

Thus

~

19//~ ,

f o6

e = 6

is a

~

Ws

onto a

.

By

6

E Hom(9/,~)

.

[K E a N F : (c+h)FK ~ pK} =

Ws

, so

(c+h) r E aN(P)

with unit element

together with a function

such that

is

Since the functian

[K E ~ ~ F :c(K,hK) F ~ pK] = [K E ~ ~ r :hK ~ ~qK} iff

f

~

1.3.1,

6

with unit element

mapping -I

Y

~X (~~

one-to-one

induces an isomorphism ~Y(q)

.

Thus

t of =6

is the desired homemorphlsmo If

IUI < w ,

then

IU N Rgql = IU 1 ~ IAI . a

Ws a

IUI = IX1 By

with unit element

and hence

1.3.18(ii), ~W (q)

t'f*6*9/

for some

W

function

s oq = q .

induces an isomorphism

onto a

Ws

1.3.1,

s

with unit element

~U (q)

mapping

IUI = IW I ,

W

and hence

Assume now

is ext-isomorphic

with

Thus there is a one-to-one By

s

Y = U.

onto u

U

from

to

U c W .

such that t'f*6*9/

u ot of o6

is

the desired homomorphism for the last part of the theorem.

Our next theorem, due to Henkin and Monk, Given a a

Cs

gws

is related to 1.7.21.

with all subbases finite, it is not always isomorphic

; see 1.6.8(5).

It is natural, however,

to

to try to reduce the

number of subbases.

Theorem 1.7.28. with unit element

V

Let

K < ~ ~ a 9

Suppose that

such that every subbase of

9/

9/

is a

is of power

Gws K 9

116

Ultraproducts

Let R

be the least cardinal

k

such that for each equivalence

having all equivalence

on

1.7.28

classes

infinite

relation

the following

inequality

holds:

I[q 6 V : q / q -1 = R]/ ~ k . / [ q Then

is isomorphic

9/

Let

Proof.

~U~pl) N

9/

~U~ pj) = 0

[qlq -I :q 6 ~K] infinite]

.

I[ ~ :~

(I)

m

~

< = and q~ ~ qS~] I < ~

R l ~ R"

, say

R" = rHlr "-I

since

Thus

a

if

of

q,q',rt,r # 6 ~K

function

(2)

Now set

k

of R are

by setting

R ~ R'

R t = q11qt-I

R .

R = qlq -I ,

q 6 ~K (ql)

R = (k ~

R =

classes

is questionable.

and

from

R

where

.

-=

and

on

R = qlq -I

relation on

only the transitivity

is a one-to-one

~

such that

Indeed,

R ~ R l' .

subbases.

= [R 6 ~ :all equivalence

q,ql 6 ~K

K < w 9

k

= ~ i61 ~Ui (pi)

V

i,j 6 1 .

Now we define a relation

,

with

have unit element

is an equivalence

(rtlr l-l)

Gs

for distinct

and

iff there exist

and

to a

6 ffK : q l q "1 = R] / 9

K

onto

~

,

Suppose

R =

R' = (qllq1"l)

r u 6 ~K (rl)

K

such that

and

k~ q 6

=

Then there k ~ q' = r' ,

~ (r")

, so

Similarly we have

R =- R s

q' 6 ~U (q)

,

Iul

=

such that

The following

statement

(3> if R6

=,

IUI

q6

u,

and

R = qlq-I

, then there is

R l = q'lq '-I .

is clear.

,and

q 6 ~U

, then there is at most one

1.7.28

q

t

Ultraproducts

( ~ E ~U "q"

(4)

such that

For every

For, say

R E e

Since

i-1

= R .

there is an

R = qlq -I , where

I~/RI < ~] Choose

q'lq

K < w

such that

R' 6 @o~

q E ~K . we have

~ E ~'~ F , and let

~17

Let

R--- R' .

F = U [ B / R : 8 < ~ and

I~/R 1 < •

and hence

IF[ < ~ .

q' = (~ ~ F)lq U ;

q'

is

as desired. Now by -I pilp i = R i

(I) - (4) for all

Now let each of power is a

~ < k

i E I , and


there is a function

< k>

Set

such that

q 6 ~YB

with

qlq -I = Ri] .

(5)

if

i,j E I

and

(6)

if

i E I , then

i 6 1

.

and

W , and the

Next, for each

whenever

such that pilPi I = pjlpi I

be a system of pairwise disjoint sets,

W = U~
equivalence relation on K = W/N .

R i = R. 3

R E I~

We define

q' 6 ~Y~q)

Thus

iff there N

is an

~ -classes are weak spaces.

let

L i = IS 6 K : there is a

Then by the choice of

R i ~ Rj ,

q ~ q'

then

Let

qES

R ,

L i N Lj = 0~

li/(RIR-l)l ~ ILil

In fact, using finally the hypothesis of the theorem, and (3),

Ii/(RIR-I)I = l[q 6 V : qlq -I = Ri] 1 K l[q E W : qlq -1 = Ri] 1 = ILil By

(6)

we can choose a one-to-one

all

i E I .

Then

(7)

for all

i s I

there is a

S E IK

q E Si

such that

such that

ql q

S. 6 L. l i

-I

for

-I = pilp i

118

Ultraproducts

This is true since

pilp ~I = R i

onto

mSi = i

I

such that By

1.6.2

unit element the full to a

Ws

Gs

we have

~.(pi) -i

1.7.28

S i E L.I " Now let

and

for all

i E I .

with unit element

with unit element

map

K

is a

Ws~

with

i E I .

~ 91 = PiEI ~i ' where

for each

m

~i

For each

T . By 1.6.2

W , which has

let

PTEK ~T

k

to prove the theorem it suffices to show that

T s K

~T

be

is isomorphic

subbases.

Thus

PiEI ~i ~ I =

PT6K ~T "

First we note:

(8)

for e~ery

For, say

~l

where

(9)

~

that

1.7.27

(7)

i 6 1

Y8

Then

~ oh

such that

1.3.1 .

By

(8)

and

(9)

E Hom(~mT,~ T)

~mT

~T "

onto a

is as desired in

~i

into

klk -I = pilp~.I

There is a ore-to-one function

5 o Pi = k . Thus

and

i E I .

be a homomorphism from

~U mT (r) "

k E Si

1.3.5

~

into

(8) '

1.3.5.

~ < k 9

such that

h

~mT

be one-to-one and onto, and set

there is an isomorphism from

choose

for some

b E UmTy~ let

is defined in

For every

~y~(k)

by

By

Let

with unit element

For, by

onto

there is a homomorphism from

T = ~Y~q) ~ (

r = b-I =q . Ws

T E K

~

T E K

and

h

WSi =

from

hsi E Ism(~i,~Si)

g = < < ~ X m T :T 6 K> :x E PiEI Bi> Pi61 ~i

b

with domain

It follows easily that

is the desired isomorphism from

Thus

U.l

is the desired isomorphism,

there is a function for any

~Si "

into

PT6K ~T "

K

such for each

I. 7.29

Ultraproducts

Corollary 1.7.29. Gws

Let

Ill ~ K .

i,j 6 I .

Then

Proof. relation on

~

K < ~ ~ ~ .

Assume that

is isomorphic

K > I .

l[q E V :q[q-I = R} I < ii I , since if

q 6 ~u(Pi) ~i

q ~ q' , then q' 6 ~U j(pj)

'

~

is a

for all

i s I , and

Cs

having every equivalence

q,q' 6 V , and

that

~l,(Pi) N ~u(PJ) . = 0 i 3

IUil = K

to a

We may assume that ~

Suppose

V = H ~iEl ~U i(pi) , where

with unit element

for distinct

119

Let

R

be an equivalence

class infinite.

Then

qlq -I = R = qSlqt-I

[~ < ~ : q~ ~ q'~]

for distinct

with

~s infinite and so

i,j 6 1

Thus by

1.7.28

it

suffices to check

(*)

if

R

the set

is an equivalence

[~K (q) :qlq -I = R}

To prove

~

and

has at least

K

elements.

(*) , first choose

be the permutation while

relation on

of

K

f(K - I) = 0 .

fk o q ~ ~K(f~~

q s ~K

such that

such that fk = ~ + I

Then for all distinct

, while

I~/R 1 K K , then

qlq -I = R . for all

k,~ < K

Let

f

k < K- I ,

we clearly have

(fk o q)l(fk o q)-I = R , so

(*)

follows. l

Remarks 1.7.30. (a)

(These remarks are due to Andr~ka and Nemeti.)

The special hypotheses

~ IUI ~ ~ and any

w s UK

then there is a

~ 6 Cs

of power

in

• Hg/

> ~ .

1.7.19

Ws

such a

be one-to-one and onto. Ws

a 6 KC .

For every

~/ with base

U

Namely,

such that

if

IAI ~

either has only one element or else has base

To construct

the full

are necessary.

with unit element k < K

let

Let

Ws

, let

Let

p = <0 :~ < ~> , and let

V = ~U (p) . kk

IU[ = K .

~

be

We now construct

be a one-to-one

function mapping

120

K

Ultraproducts

into

{v E U : X < wv}

Then for each

a x = ~q ~ V : ( k m e x { w q ~ Now i f

0 < ~ < ~

wq~ ~; m x

so

(1)

~

qo

~ if

"

,

:0<~

k < K , and

{wq, 0 : 0

I. 7.30

k < K

<~])X

let

= qo]

"

q E a X , then

< "o < o/] < w ( k max{~wq~ : 0

< ~ ] ) ~ . = WqO ,

Thus

0 < ~ < ~

and

X < K , then

a x = -do~

-

We also clearly have

=

(2)

Coa X

I

for all

(3)

ax N a

Let

9/ = ~ ( ~ ) { a x : X < K} .

(4)

if

= 0

for distinct

~ 6 Ws

n Hg/

For, suppose that conditions For each fa x

Thus

YX ~ Rgq

f

(I) - (3) X < K y for

X < K X,~ < K .

Clearly

IAi g ~ .

with unit element

Now we claim

~Y(q) , then

is a homomorphism from

9/ onto

hold with

replaced by

choose

(ak : k < K)

YX 6 Y

0 < X < K

by

(3)

(I)

.

Thus the .

0 qYX 6 fa x , using

so that

is one-to-one by

~

Iy N Rgql z K 9

for

fa x

for fa x , if

and

fa~ .

YX ~ q0 .

(2)

for

Furthermore, So

(4)

holds. Now suppose and assume that Then

h 6 Hom(9/,~)

IWI ~ ~ .

r2~ o h 6 Hom(9/~)

contradicting

(4) .

for some

Let

q

map

, where

B

is a

Cscr ~

ff onto Ws

with base W,

and set

W ~ 0 , V = ~W (q)

with unit element

V ,

1.7.30

(b)

Ultraproducts

In

1.7.27

In fact, let [p} , where Ws

one cannot replace '~omomorphic" by "isomorphic".

~

be the

Ws

U = ~ + ~

with unit elermnt

and

with unit element Since

r E f[p]

r E dKk

~U (p)

p = <0 :K < 5> , and let

~U (q) , where

f E Ism(~,~) . with

72~

[p] ~ dKk

~

be the full

q = .

for all

for all

generated by

Suppose

K,~ < ~ , there is an

K,k < ~ .

Since

r E ~U (q) , this

is impossible. (C) in

By the argument for 1.7.27

(a)(4) above, the condition

is necessary.

(d)

It is not known if the condition

(e)

The hypothesis on

Namely, suppose that every subhase of Gs

~

with

k

~

k ~

in

IUl ~ IAI

in 1.7.27 is needed.

1.7.28 is in a sense best possible.

is a

@ws

is of power

with unit element K , and that

~

W

subbases; assume in addition that

relation on

Be the unit element of ~

Let

R

E R

for each


and

plp -I = R

for all

P E h[q]

p E h[q} p E h[q]

implies that ; and similarly So

q E V Thus

and

such that

[q] 6 A Let

for all h E Is(~,~) ,

be an equivalence

with all equivalence classes infinite.

qlq -I = R , then PK = Pk

~ .

V

is isomorphic to a

q 6 V ; we show that the indicated inequality holds. and let

IU ~ Rgql = IUI

[q] ~ ~

If

q E V

and hence

pK ~ pk

whenever

qlq "I = R

imply



system of pairwise disjoint non-empty subsets of

that is a

[q 6 W : qlq -I = R] .

Hence

l [ q E V : q / q -1 - R} ~ I[q E W : q l q -1 = R]I --k'l[q E ~

and

: qlq "1 = R] I 9

122

(f)

Reducts

The assumption

upon considering (g)

III ~ K

1.8.1

in

1.7.29

cannot be improved, by

R = ff X ~ .

Andr~ka ar~ N~meti have proved the following algebraic version

of the various logical

theorems to the effect that elementarily equiva-

lent structures have isomorphic elementary extensions: and

(e)

~ ~

.

Then

respectively

~

and

such that

~

Ot

are sub-isomorphic

and

O'

8.

We restrict ourselves

to

Let Cs's

Ot

E Cs and

~'

are base-isomorphic.

Reducts

in this section to the most basic results t

about reduets.

~

t

A more detailed study is found in Andreka, Nemeti

[AN3]

to which we also refer for the statement of various open questions.

Lemma 1.8.1. V .

Let

~

For each

Y

+

y+~Su. Then

f

Proof.

be a

Crs~

f

Y E ~U

For all

Y~A

W = fV . x~

preserves

and unit element

be one-to-one.

Fix

p-l)

let m--~y~=U:y+~Y~ ~

Clearly

(p)91 into a

f

preserves

6 fX , i.e.,

fX ~ 0 .

dKk

U

set

is a homomorphism of

Let

with base

p E ~

= ((6 N Rgp)Ix) U (Y ~

( x o p ) + = x , we have check that

~

be an ordinal and let

x E X E A .

thus

Let

for

K,k < ~ 9

Crs

+

, and

and

fX ~ 0 .

Sir= e

It is routine to

Now suppose that

Y 6 A ,

1.8.2

123

Reducts

K < ~ , and

y EW 9

fc

zz , let

~ ~K

Y+ 6 c[V]ypK "

Thus

Y Efc

y+ s V

easily checked that and so

For brevity set

)Y . Thus

and

The other inclusion

Theorem 1.8.2,

If

~

and

~

,

for some

,

u E U .

K + 6 Y , so (yu) =

It is

YuK E fY

is established similarly.

are ordinals with

Rd ~P)GwsB ~ I G w s

is one-to-one, then

To prove that

y E fC

. +.pK iY )u 6 Y

(y~)+ = [Y . +,pK )u ; hence

y 6 c~W]fY .

p E~

~ = ~(P)9/ .

, and

~ m 2

Rd (p)

and

GWS~

I Gws

Proof.

First we take any

9/ E Ws~

and show that

~

(p)

9/ E I G w s

To this end, by 2.4.39 and 1.6.4 it suffices to take any non-zero

X E A

and find a homomorphism

fX ~ 0 .

f

of

~(P)~

into some

Ws

such that

Say 9/ has unit element ~U (p) , and x E X . For each y 6 ~U define + y as in 1.8.1; then define f as there also. Applying 1.8.1, we see that

f

is a homomorphism of

~ (p)9/

Now it is easily checked that Fy = [K 6 Rgp :yp-IK ~ pK} Ip -I* Fy I < ~ , and ~U (p=p) , so

~

into a

Crs

f(BU (p)) = [y 6 ~U

for all

y s ~U .

Ws

fX ~ 0 .

:Iryl < |

, where

Clearly

p-l*Fy = [K < ~ :yK ~ ppK] .

is a

~ , and

Thus

IFyl < ~

iff

f(~U (p)) =

as desired.

,

The theorem itself now follows easily from 0.5.13(iv) and 1.6.4, the firml statement being clear from the above.

Remarks 1.8.3. we also have and

Rd (p)~ C s

Under the hypothesis of 1.8.2 and using 1o8.2

~ IGs ~ R d(P)Gs8 ~ = l~Cs~if

have shown that if

Rgp ? ~

and

~ ~ ~ then

g e n e r a l i z i n g examples of Monk.

Rd(P)Gsreg ~ 8 -~ l~ u s reg by 1.7.21 .

by

1.7.14,

l

But Andr~ka and Nemeti

Rd~ Cs~ ~ ICs~

and Rd~O)Ws~ ~ I W s

124

Reducts

Theorem 1.8.4. one-to-one and onto. Cs,Ws]

Let

~

Then

y

is a

The arguments

1.8.1

= IK

for

K = Crs

-I

The hypotheses of

fX ~ 0

~(P)9~

K E [Crs,Gws,Gs, Gws reg,

K = Gws r e g .

and unit element

into a

Crs

~

f

y+ =

defined there

with unit element f

fV ,

is one-to-one.

Thus

It is easy to check that

Y E A , y E fY ,

For any

c~)Y

K < ~

we have

c~)fY

~(~)fY = p-l*~(9/)y .

p'lo E ~(~)fY , and so

= fY

iff

First suppose that

Clearly

by the assumed regularity, so

a(~)y

.

Now

= Y

iff

pK

+ 0

(Y)yO

' so

c_ IK 8

Second,

(using

+0 (z)zO

E V , and

E Y 9

Hence

in our two representative cases.

For the other inclusion, it suffices to note that Rd~ p- I ) K

z+ E V , so

E V

+0 (z)yO

Then

Hence

as desired.

Clearly also

+0 +0 (A (9/)Y U 1)I ( y ) y O = (~(9/)y U 1)I ( z ) y O + z ~ Y and z E f Y , as desired. IRd(P)K~ _c I K

so

and

~ = ~&(P)9/ .

c(9/)Y = Y .

O s ~(9/)y 9

y+ E Y

z E fY

Y+ E Y ,

+0 (Y)yO E Y .

~/ E Gws~) , and hence

We have shown

Now let

yp-lo = zp-lo , i.e., y+O = z+O .

(gCg/)Y LJ I)IY+ = (~(91)y U l)Iz + 9

O ~

((~(B)fY) U l)ly

z E W , and z E fY 9

one and onto,

Now the function

x ; we have simply

X E A , i.e.,

9/ E Gws r e g .

Assume that

assume that

First suppose

V .

= ((~(~)fY) U l)Iz ; we want to show that

+ z E Y

be

, as desired.

Now suppose that

Thus

~ 6 ~B

1.8.1 hold, so the function

for every non-zero

~(P)9/ E ICrs

E Gws

U

and

does not depend on any element

is a homomorphism of and

be ordinals and let

being very easy, we restrict ourselves to

Crs B , say with base

in

y ~p

B

IRd (p) K

two representative cases,

+

and

.

Proof.

9/

1.8.4

p

-i

E

B~

is one-to-

by what was already shown, and clearly

1.8.5 Rd(P)Rd(p-l)K

= K

N o w we turn to neat technical

.

embeddings,

for w h i c h we a l s o

require

a

lenmm.

Lemma V

125

Reducts

1.8.5.

Assume

that

Let

~

~ ~ 8

be a

and

Crs

W ~ ~U

with base

.

U

and unit e l e m e n t

We also assume

the following

conditions: (i)

V = {x : x = ~ l y for some y E W]

(ii) K x u EW

for all

Then

XEA

there

preserves

~x

6 V

x E W N

Let +

Crs~ ~

u E U , if

and

~

.

c~)fX

be t ~ f u l l

N o w let

x E W

Hence

f

preserves

(i)

there

is a

y E W

shows

x E f(V N X)

that

f

~Ix K u E V

then

; we w a n t x 6 W u 6 U

xKu 6 W

.

Finally, x 6 W

and

.

(i)).

to show that and

CrsB

w i t h unit element

and

Hence suppose

X E A

-

If

such that

.

suppose

9 ,

x uK 6 fX

for some

f

.

preserves X E A

fx .

,

.

Clearly

iff x s X .

Y E fX dKk

K < ~

.

for ,

and

By the definition

~Ix E V

K 6 B N ~

such

we have

Thus

and

our a s s u m p t i o n

u 6 U

.

, choose

x 6 C~ W] fX , as desired. ,

W

0 # X E A x ~ y

Hence

(i)

W

K 6 8 " ~

~Ix ~ X

that

x ~ c~W~ K

X E A

By

and

iff

Clearly

(~Ix)Ku 6 V

that

X s A

~Ix E V N X

~ I x E C~ V ] X

Since

a n d unit element

for all

iff

Now

U

.

= fX

is one-to-one.

(again u s i n g

for some

w i t h base

9

we have

Thus

is a

fX

x E C ~V]x

that

, and

fX = Ix E W : ~ I x E X]

Hence

K,k < ~

f

K < ~

o

T h e n by This

let

f 6 Ism(9/,~)

Proof. f

,

. For any

that

x E W

;

(~IX)Ku E X

(ii)

The converse , and

Hence

yields is similar.

_[W] fX x 6 ~K

x uK E W

of

and

9

126

Problems

~Ix K E X .

Since

1.8.6

~Ix E X , and hence

K ~ ~ , this means that

U

x E fX , as desired.

From this l e ~

it is easy to prove

Theorem 1.8.6. Then

K

~ ISNr ~

Assume that

~ ~ ~

and

K E [Ws,Cs,Gws,Gs}

9

.

Corollary 1.8.7.

If

2 ~ ~ ~ B

then

IGws

= SNr IGws B =

SNr IGs B = I G s

Proof.

By

1.7.14, 1.8.2, and 1.8.6.

Remark 1.8.8. ordinal

~

we have

It follows from 2.6.48 and 1.8.6 that for any Cs

U Ws~ U Gs~ U Gws~ c SNr Dc0t~

result of the representation paper, is that if

~ ~ 2

theory of

then

CA's

9.

A major

, to appear in a later

= IGs

SNr Dcct~

.

=IOws

.

Problems

We begin by indicating the status of the problems listed in [ HMT ] as of January 1981.

In Problem 0.6 one should assume that

than the first uncountable measurable cardinal Under this corrected

formulation,

relative to the consistency shown by Magidor

of

[Ma] and Laver

as less

(see Chang, Keisler [CK] ).

the consistency of a positive answer

ZFC

plus certain other axioms has been

[L ]

!

affirmatively by B. Sobocinski

~

Problem 1.2 has been solved l

[S ]

Andr~ka and Nemeti solved

Problem 1

Problems

Problem 2.3 affirmatively;

see

127

[AN~

solved affirmatively by J. Ketonen hence for discrete D. Myers

[M~

CA's.

and

[N ].

[K ]

Problem 2.4 has been

for Boolean algebras,

and

Problem 2.8 was solved affirmatively by

and Problem 2.9 negatively by W. Hanf

[ H ].

2.11 was

J

solved negatively and

[ N ].

(except for

~ < 2) by Andr~ka and Nemeti;

see

Problem 2.12 was solved negatively by R. Maddux

[Md].

Now we shall list some problems left open concerning

Problem i. is there a

Let

~ ~ w 9

Given a normal

Cs~ 9/ with same base

U

Gws

~

[AN2]

set algebras.

with base

U ,

such that

Io2.6-Io2.13).

Problem 2.

Let

q

be the function defined in 1.4.8.

For every

+ 6 w N 2

let

q ~

be the largest

8 s w

such that

q(~,B) = I . +

Give a simple arithmetic

Problem 3. (Cf 9

Is

description of

IW~

q , or at least of

q

.

closed under directed unions for

1.4.8 and 1.7.11.)

Problem 4.

Is

Problem 5.

Does

Problem 6.

ICs

= HCs reg

I Cs

= H~ W s

Is

Icsreg=

HPWs

Problem 7.

Is

H=Ws

Problem 8.

Is

HP=Ws

HP=Ws

or

~.

or

._ reg HCs ~ = n~ss ~9

(Cf. 1.5.6.)

(Cf. 1.5 6(17) and 1.5.8.)

ICs

?

ImCs~ ?

For these two questions cf. 1.6.8 and 1.6.10.

= HPWs

?

(Cf.

1.6.8.)

128

References

Problem 9.

Is every weakly subdirectly indecomposable

morphic to a regular

Problem I0. (or

Problem 9

Cs

Cs

iso-

?

Is every weakly subdirectly indecomposable

Cs reg) isomorphic to a

Ws

Gws

?

For these two questions cf. 1.6.16. Problem Ii.

Is the condition

IUI ~ IAI

in

11.7.27 needed?

(Cf. here also 1.7.30.)

REFERENCES

[ANI] Andreka, ' H. and Nemetl, ' " I., A simple~ purely algebraic proof of the completeness of some first order lo$ics, Alg. Univ. 5(1975), 8-15. !

[AN2] Andr~ka, H. and Nemeti, I., On problems in cylindric algebra theory, Abstracts Amer. Math. Soc. 1(1980), 588. l

l

[AN3] Andreka, H. and Nemeti, I., On cylindric-relativized this vol.,

set algebras,

[AN4] Andr~ka, H. and N~meti, I., Finite cylindric algebras generated by a single element, Finite algebra and multivalved logic (Proc. Coll. Szeged), l

!

eds. B. Csakany, I. Rosenberg, Colloq. Math. Soc. J. Bolyai vol. 28, North-Holland, to appear. [CK] Chang, C.C. and Keisler, H.J., Model theory (second edition), North-Holland 1978, xil + 554 pp. [D] Daigneault, A., On automorphisms of polyadic algebras, Trans. Amer. Math. Soc. 112(1964), 84-130. [De] Demaree, D., Studies in algebraic logic, Doctoral Dissertation, Univ. of Calif., Berkeley 1970, 96pp. [EFL] Erd~s, P., Faber, V. and Larson, J., Sets of natural numbers of positive density and cylindric set algebras of dimension 2, to appear,Alg. Univ.

References

129

[H] Hanf, W., The Boolean algebra of logic, Bull. Amer. Math. Soc. 8(1975), 587-589. [HM] Henkin, L. and Monks J.D., Cylindric set algebras and related structures, Proc. of the Tarski Symposium, Proc. Symp. Pure Math. 25(1974), Amer. Math. Soc., 105-121. [HMT] Henkin, L., Monk, J.D., and Tarski, A., Cylindric Algebras, Part I, North-Holland (1971), 508pp. [HR] Henkin, L. and Resek, D., Relativization of cylindric algebras, Fund. Math. 82(1975), 363-383. [HT] Henkin, L. and Tarski, A., Cylindric algebras, Lattice theory, Proc. Symp. pure math. 2(1961), Amer. Math~ Soc., 83-113. [K] Ketonen, J., The structure of countable Boolean algebras, Ann. Math. 108(1978), 41-89. [Ko] Koppelberg, S., Homomorphic images of Proco Amer. Math. Soc. 51(1975), 171-175. [L]

~ - complete Boolean algebras,

Laver, R., Saturated ideals and nonregu~r ultrafilters,

to appear.

[aM] Magidor, M., On the existence of nonregular ultrafilters and the cardinality of ultrapowers, Trans. Amer. Math. Soc.249 (1979),97-111 . [MI] Monk, J.D., Singularu cylindric and polyadic equality algebras, Trans. Amer. Math. Soco 112(1964), 185-205o [M2] Monk, J.D., Model-theoretic methods and results in the theory of cylindric algebras, The Theory of Models, Proc. 1963 Symp., North- Holland, 238-250~ [Md] Maddux, Ro, Relatio~ algebras and neat embeddings of cylindric algebras, Notices Amero Math~ Soc. 24(1977), A-2980 [My] Myers, Do, Cylindric algebras of first-order languages, Trans. Amer. Math. Soe. 216(1976), 189-202. I

IN] Nemeti, I., Connections between cylindric algebras and initial algebra semantics of CF languages, Mathematical logic in computer science, eds. B DSmSlki, T. ~ergely, Colloq. Math. Soc. J~ Bolyai, vol. 26 North-Holland (1981), 561-606 . IS] Soboclnskl, B., Solution to the problem concerning the Boolean bases for cylindric algebras, Notre Dame J. Formal Logic 13(1972), 529-545~ [TV] Tarski, A. and Vaught, R.L., Arithmetical extensions of relational systems, Compos. Math. 13(1957), 81-102.

On cylindric-relativized

by

This theory

is b a s e d

of cylindric

[HMT3. are

work

Most

H. A n d r 6 k a

on the book

algebras

cylindric-relativized

voted

to t h e

study

classes

of Crs-s

Gs reg.

The

played the

fundamental

the

introduction

ting

classical

in a s e n s e ture

of

of

tion

t h i s way.

tion

to

by

first

proved Gs r e g

much

order

in

[G]

Following

these

use

the n o t a t i o n s

is a c o n t i n u a t i o n

[HMTI].

refer

to

We

individual

[HMTI3. first

[HMTI32.2

The

in t h i s paper

this paper,

i t e m of

[HMTI3

sections;

items

figure

the present of

For

is

is role

example,

Gs reg,

the c l a s s

given

see

connec-

GsregnLf,

to t h e m e t a - s t r u c -

interpretations

structure

we

class

is e x a c t l y

all

considerable

introduced

paper

The

and

motivations

The present

as

that

was

is d e -

c a n be r e p r e s e n t e d

insight

shall

be-

and

give

simplifica-

special

atten-

G s reg.

shall

of t h e

attention

this

work

to the

theory.

to C A - t h e o r y

that

achieving

is s i m i l a r

IN3

of CA-s

distinguished

and CA-theory

theories

theory

a distinguished

in

The abstract in t h e b o o k

The present

algebra

proved

theory

developed

to c e r t a i n

Such

theory

It w a s

Recently

all

(Crs-s).

in B o o l e a n

[HMTI3.

abstract

in C A - t h e o r y

model

model

for t h e

precisely

Gs reg

[HMTI3.

It w a s

isomorphically

by

finitary

is e x t e n s i v e l y

[HMTI].

algebras

at least.

them.

We

set

in

I. N 6 m e t i

a n d the p a p e r

examples

more

link between

consisting

tween

introduced

by Boolean

[HMTJ

set algebras

of C r s - s ,

role played

and

(CA-s)

of the motivating

set a l g e b r a s

in

paper by

moreover

i t e m 0.5.

recalling

a n d is o r g a n i z e d

we have

the

means

this

EHMTI3

without

discussion

therefore

item of

1.2.2

figures,

item

In g e n e r a l ,

practically

are n u m b e r e d

I and

strings

[HMT3

f o r an i n t r o d u c t o r y

[HMTI]

is a l w a y s

of

in

the

of

by three

we omit

figures

O.5.1

when

read

O, from

like

the

refer

refers

in s e c t i o n

to

contents

section-titles

e.g.

We

parallel

of t h e

figures

it,

[HMTI3.

e.g.

is f o u n d

same

them.

1.2.2.

reference

to i t e m s

of

to i t e m 0 . 5 . 1

a n d it is a s u b left

to r i g h t

132

correspond We

to the

shall

proved

subdivisions

be g l a d

(or not

to send

proved

Acknowledgement. guiding logic

us

full

work

of

grateful

statements

as in o u r

work,

research

to is found.

claimed

but

whenever

to P r o f e s s o r

J.D.

not

requested.

Monk

concerning

for

algebraic

in g e n e r a l .

O. B a s i c c o n c e ~ t _ s _ a n d

We use

the n o t a t i o n s

recalling Ws,

item referred

in the p r e s e n t

are m o s t as w e l l

the

proofs

in detail) We

in this

in w h i c h

Gs

them. , Gws

-relativized normal

set

we use

, C s ~ eg,

algebras

[HMTI3

EHMTIJl. I

G s ~ eg,

were

of

and

where

the

Gws reg~ , Crs rega

introduced.

E HMT3

without

classes

Cs

,

of c y l i n d r i c -

All

these

algebras

to

CA -s.

are

Bo -s.

Notations:

Let

%%

exists

since

Mn(~)

~ Sg(~){l

Let unit

and d e f i n i t i o n s

Especially , Crs

notations

V

i

is a c o n s t a n t ~

be a

V.

This

be an a l g e b r a

}

Crs

and

notation

symbol

~(~)

-unit.

similar of

~

Then

CA -s.

~(4A){I~

~V

is a m b i g u o u s

i

We d e f i n e

}.

denotes

if

Then

V=O

the

full

but we h o p e

Crs

with

context

will

help. Let

xCVC~U.

Then

AEVJx

~

{ie~

: c~V3x#x}

and

A(U)x

~ AE~U]x.

1

of

Let

H

be any

H

and

Gc

H

set.

Then

denotes

Sb H that

denotes GESb

--W

As

the

set of

all

finite

subsets

H. W

a generalization

of the

notation

f~

introduced

in

EHMTI],

U

the

following

and

let The

H

be a set.

notations

f : A >~ one-one

notation

B into,

mean

will

be v e r y

Then

fEH/k]

f : A ~ B, that

one-one

A1f onto

f

useful. ~

(Dof

: A ~-- B,

is a f u n c t i o n respectively)

~

f

Let

f,k

H)If

U H1k.

: A >-- B,

mapping B.

be two

A

functions

and into

In a c c o r d a n c e

(onto, with

EHMT3,

0.1

133

fEIs(~,~)

means

We

use

shall

[HMT~

to

Crs

not be a

CA

A(~)x

{ {iE~

~ By of

-s

as w e l l ,

E.g. : c~x 1

2.2.3

1.1.1

in t h e i r

true

>

-s,

for

Crs

the

-s,

of

apply

1.2.1-1.2.12

1.6.2, Let

1.6.5-1.6.7 ~,~ECrs

Zd A ~ Z d ~ will

Boolean

holds

V

we

Subu(V)

the

of

we

U

let

At

be a n y

an a t o m

of

set of a l l

too.

the B o o l e a n subunits

of

~ u{Rgp

subbases

. Then

base(~)

~ base(l~),

subunit

of

, 1~ .

and

Y = base(W)

is s a i d Y

is

1.6.5-1.6.7

( C o ) - ( C 3)

to

we

sets.

shall

-s.

By

let

applicable of

and

we

Crs

Therefore

to

Cr

The

Particularly

for of

V~U

-s

above

for a n y

By a s u b u n i t

field

Zd Sb V

V

I.e.

: pCV}. some

We

of

sets.

Subu(V)

=

say t h a t

Y

W ~ Subu(V).

V Subu(~)

to b e a s u b u n i t

said

1.6.2,

~CCrs

let

set o f all

W

(as it

B ~ Ate.

the

= Subb(l~).

(Co)-(C3)

every

fields

set a n d

base(V)

only,

in the p r o o f s

notions

set a l g e b r a s

CA

use only

~=/~

iff

Subb(~)

of

implies

(C 7)

-s

EHMT]

for

V

denotes

a subbase

~

We define

1~

~CBA

need

d Ax =

1.2.1-1.2.12

for

1.6.5-1.6.7

In g e n e r a l ,

Let

understand

~J~ECrs

that

A=B

for Boolean

Zd Sb V

Let

1.6.2

Crs

(C 5) a n d

EHMT]

are

the a b o v e ,

to c y l i n d r i c - r e l a t i v i z e d

denotes

Subb(V)

of

proofs

a

C A -s in

2.6.18.

stated used

for

and

(C0)-(C3),

EHMT3) . A l s o

and

~ECrs

axioms

their

Because

Then

O.1.

is a s u b b a s e

of

and

etc.

Then

: O}

Therefore are

that

xEA.

: A(~)x

axioms .

fact

let

1.6.1

they

only

we have

set a l g e b r a

Definition

= At

for

be applied

argument

of

.

Crs

since

1.2.2-1.2.12 EHMT~

in

on p.177

EHMT].

[HMT]

for Hom

introduced

the

and

{xeA

the

etc.

of

,

Zd~

that

similarly

Zd,

despite

although

proofs

noted

A,

cf.

we have

Crs

and

~ECrs

are v a l i d

for

is e x p l i c i t l y

let ~ x},

EHMT]

because

are

notations

+,',-,O,1

true

A1fCIs(~,~)

the

( Zd~,

[HMT3

are

that

of

to be a s u b b a s e

= Subu(l~)

and

~

is a

of

iff ~

W iff

Y

is

134

0.2

The that

above

a subbase

Notation:

Let

subbase

be e m p t y

KCCrs

and

iff

~

~ {~eK

: (VUeSubb(~))IUlk~}

that

notation

agrees

the one-element

with

Crs

agrees e:O

with

EHMTI3

i.i

(vii).

Note

.

be a c a r d i n a l .

: (VUESubb(~))IUI=x}

above

has

might

of

K ~ {~EK K The

definition

Then

and

EHMTI3

is in

5.6(17)

and

Crs N Crs

EHMTI3

for a l l

7.20.

Note

x since

it

no s u b b a s e s . In t h i s

I•

That

=

{~eCAa

is

Lemma

0.2.

recall

is a v a r i e t y

consisting

EHMT]

we

of

2.6.54

Let

and

EHMTI]

$J[eCrs

Then

is t h e d i s j o i n t

ii)

Let

WeSubu(~). some

~EGws

Moreover,

=

O<•

Then

and

the a b o v e

scheme

one.

of

This

is i m m e d -

8.8.

(i)-(•

union

Then

by a f i n i t e

below

of a l l

W C e y (p)

hold.

subunits

for

some

of YCSubb(~)

and

pEI ~ iff

that

Let

~+1)•

(C0)-(C7)

1O

iii)

following:

definable

i)

for

the

: ~}g~ (c(~)~(~•

I Gs

equations iate by

connection

every

let

~k2

(Vi,jEm)Ei#j

{~y!pi)

subunit and

of

let

~

is a w e a k

V : U { e Y i(pi)

~ ey!pi)neY!PJ) m ]

: O]

space.

: i6I} Then

be

such

Subu(V)

=

iel}

:

I

The proof

Lemma

of L e m m a

0.2.1.

Let

< ~{Cio...CinX (i) ~

zd

0.2

depends

~

be a c o m p l e t e

: new,

: A ~ Zd~

,

on the

iEn+l~} zd

: At~

following

SCr

: xeA > . ~ At

Zd~J[,

lemma.

Let Then and

zd (i)-(ii) (VxEA)

below zd(x)

hold. :

135

0.2.1.

= H{yEZd~ closure" (ii)

If

Proof.

as

Proof Zd0~

2~

and

i E

n+l

x
We

Then

zd(x)-y

then

zd(x)

Suppose

x~y

by

and .

. Then and

therefore

of 0.2.1(ii) : follows zd

zd

: A

zd(x)~y

1.6.7. by

y = zd(x)

Suppose

~J~ is a t o m i c

and

At~

of B o o l e a n

and by

that

i~

(Co))

y E Zd6~.

x~zd(x)-y

and

x~y E Zd~

zd

theory

new

y ~ zd(x).

Therefore

the

If

x S zd(x) by

.

Suppose

x~zd(x),

c At~.

proves

be

xeA

(by

: x~y)

y=0 by

Therefore

from

: At~

S ~ At ~J~

x ~ -y

Then

zd(x)

zd

for e v e r y

zd(x)~y

Let and

that

for a n y

Cio...CinX~Y

therefore

Then

: x~y}

seen that

function

: x~y}

1.6.6

= H{y E Z d ~

QED

= H{y e ZdD5

EHMT3

,

the

it is c l e a r

c At~

zd(x)

= ~ At~

1.2.6(i)

zd(x)

x E Ate.

by

let

show that

since

At~2~,

[HMT3

1.2.7

~ zd(x)-y,

= ~ At~

and

= Z{y E Zd~

x ~ zd(x)-y

Proof

SCr

zd(x)

6 Zd~

"zero-dimensional

= ~ At ~OL.

then

x~y

i~

EHMT3

~ Zd{]~

x E A t O~

i~

zd(x)

suppose

by

is t h e

of 0 . 2 . 1 .

Using

show

and ~

then

statement

Next we

y E Zd~

zd(x)

.

of O . 2 . 1 ( i ) :

.

We have

x

i.e.

be a complete

in t h e

Let

By

of

is a t o m i c

Let

defined

: x~y},

by

therefore and by

y~zd(x).

: At~

At~/A.

complete.

algebras.

(~xEA)x~zd(x).

Then ~ At~

Then

1~ =

= ~ At ~ .

(Lemma 0.2.1.)

Now we

turn

to the p r o o f

of L e m m a

0.2.

Let

05ECrs

and

let

~= gel ~ Proof 1~

= iI Proof

We have

of 0 . 2 ( 1 ) : = ~ At

~

is a c o m p l e t e

Zd~ = ~ Subu(~)

of 0.2(ii) : Let

p E W E At

Zd Sb

1~

.

: new,

Therefore

SCr

,

hence

by 0.2.1.

W e Subu(~)

W = ~ { C i o . . . C i n { p)

and a t o m i c

and

i E n+1

let }

peW

be

arbitrary.

by O.2.1(i),

W c a b a s e ( W ) (p)

since

136

0.3

Proof

of O . 2 ( i i i ) : If e v e r y

6%c

Gws

by 0.2(i)

and by

the

other

direction

holds

Gws

-unit

is

an

Let

V : u { ~ Y (pi). : iEI}

subunit

of

4~

the d e f i n i t i o n too,

since

a -dimensional

be

such

by

EHMTI~

of

every

weak

Gws

then.

(Vi,

space

then

a~l

then

. If

nonempty

space

that

is a w e a k

subset

of

Suppose

kEI)Ci~k

a

a~2

.

~

1

~ Y !pP ki ))A ~=Y O~ ~ ' 1 =

Then

1.12 we have

At

Zd Sb V :

: iEI}.

{~y!pi) l

QED(Lemma

0.2.)

Proposition

0.3.

-(iv)

below

are

(i)

V

(ii)

The

(iii)

The

is a

Let

V

be a

Crs

-unit.

Then

statements

(i)-

equivalent.

Gws

full

-unit.

Crs

Crs

with

unit

unit

V

with

V

is

and

a

CA

generated

{{q}

by

: qEV}

is a

CA (iv)

Every

Proof.

Since

(iv) =

Gws

(i).

We

O.3.1.

Then

with

(ii) =

implies

Lemma

Crs

Proof.

Assume

algebra

of

c CA

(iii). shall

Let

b ~ s~b 3

unit

by So,

need

~CCA

holds

V

in

is a

b'ci(dij'b)=b

holds

b ~ s}b

holds

in

QED(Lemma

0.3.1.)

{76 .

1.9(i),

following

beA

and

~J~ for all

CA a in

CA

it is e n o u g h

the h y p o t h e s e s

~b ~

a

[HMTI3

the

,

is

of L e n a and

hence

it is c l e a r

to p r o v e

that

that

(i) ~

(iii)

lemma.

assume i,jEa

CA

A S~_~#

O

i

.

O.3.1. ~b~

4o~, by the d e f i n i t i o n

Then

the m i n i m a l

b cidij=l of

~b ~

.

Then

.

I.e.

sub-

0.3.

137

Now and

we

let

return

e]~ b e

Assume

to

the

Then

we

Case

show

i

,

and

enough

to p r o v e

that

i#j

Let

i,jE~

By

we

fiu : f fi( k )

{~E

CA

and

q-hqj(j) pEV

and

for

all

Case

2

and and

i,jE~

therefore

relation

> ~V on



>EV

.

and

hence

< u,w > EV ECoCl{<w,w

clc0{< w , w that

V

.

have

= q

If

are

done.

.

It

suffices

.

Suppose

>eV

by

Clearly

{{q}

: qeV}.

in

~

let

by O.3.I.

: c i ( d i j ~V) "

.

We

for

all

it

is

i,jE~

f , h E m l !3 U ) ~ v

,

such

u d = q(k).

By

V < SkV

E V

.

and

Note

fcV

and

{h

(C 4)

that

'

.

fi u'

that

Then and

{f,h}

that

{f)eA

Then

by

and

h uI} c V --

Since

"

i#j

Then

and

therefore

qEV

s jl V , s i3V

proved

c_ V

h E

h E c~c~{f}. 3 l

q(i)

have

,

=

P=fp(i)i

) cV


~_
then

Let

~:2

to

< u,v

V c _

have

< w,v

}

(<w,w >EV

}

and

> cV

: Gws

We

define

that ;

c {
< u,v

by

_< V

< u,w

just

again). .

By

and ( u,u

> ,
O.3.1 > EV

so

[HMTI]

u - v

iff

< v,u

u,vEU

) EV

.

similarly >CCoCl{
> EV

; . we

proved, Hence

these

1.8

equivalence

that

O.3.1),

was

by

is a n

prove

> } ,

by what

by Lemma

-=

we

(Lemma

suppose

therefore

Gs 2 -unit.

) cV

Crs .

show

solv

>E

Now

We

is a

by

,

are

p(i) = h ( i ) =

we

we

v,u .

-unit

and

V _< s ~ V , s J v

Let

(j)

have

Then

= fi q(i)

~<2

U

In f a c t ,

Crs

.

Assume

< u,v

we

p E ci{f }

P = fi p(i)

there

9 pCV},

(ful)3

assumption,

By

by

V < - s i3 V

in ~

hiu = h h)(ik

{{p]

hi : u

(~p e c i { f } ) h = p 3 ( j )

by

k ~ {i,j)

by

since

by our

a

holds

by

q~sl.V's3V 3 l

Then

by

c V

is g e n e r a t e d

and

k E a~{i,j} .

= q(k),

,

E V

holds

Let

Let

= f(k)

c c~.c~c~[f} 1 3 1

be

U = base(V)

V _< s 3 V

2.2

_< V

i#j

= h q(j) j

~

V

generated

Let

hence

[HMTI3

s3V's3V

,

u = h(k)

since

and

q qi( j )

= c!U)I (m i(U ) ~ V ) n C ~ U)(D(U)nv)ij .

have

Let

~EGws ~_>3

q = fiq(i)

V

~CCA

(VqEV)(Vi,jE~)

that

Prop.O.3.

unit

assume

4]~ES~

Assume

.

of

with

i.e.

In p a r t i c u l a r , Now

proof

Crs

O.3(iii),

= ~U

the

we

show

so < u,v have

> } , that

< u,v > E >C proved

138

0.4.

QED(Pr0position

Remark 0.4. with

0.3.)

In P r o p o s i t i o n

the c o n d i t i o n

by EHMTI3

Problem

2.14

Note

Let

on

V

that by

E IGs

for

~

[HMT3

2.6.57

(~CAa) b

condition

b ~ s~b's@b 3 l

E.g. :

c

A

Definition Subb(~) (VY,

0.5.

Let

~(~V)

0.3,

~

E CA

V E CA

iff

discussion

of

CA

for

~

iff

~(~V)

~V

E IGs

since

The but

in general

is a

2.2.10

3)

~b~ECA

completely,

if

,

and n e c e s s a r y

[HMT]

s~b ~ b ~ E C A

is not n e c e s s a r y

is a p a r t i t i o n

of

or

Then

~

base(n),

by

[HMTI3

as Prop.O.3 Crs

and

the 2.1

shows.

{{q)

YnZ=O3.

or

:

of all normal,

of

respectively.

above n o t i o n s 2,1,

~

is normal

iff

~]i is said to be c o m p r e s s e d distributed

iff

iff

iff

base(W)~base(V)=O3.

Then

classes

is said to be normal

i.e.

~J~ is said to be w i d e l y

K < Crs

In EHMTI3

be a

are the s u f f i c i e n t

we have

~ECrs

(VW, V E S u b u ( ~ ) ) [ W = V

The

V

.

ISubb(~) [~i.

K,

cannot be r e p l a c e d

~ CA

?

be o m i t t e d

(iii)

unit

V) e CA

is n e c e s s a r y

ZESubb(~))EY=Z

Let

What

is n e c e s s a r y

it cannot

b ~ s~b's~b

qEb}

c ICrs

the f o l l o w i n g

s~b,s~b 3 l

though

with

(VbeA)[(~i , 3 E a ) b = s ~ b

condition

2.14),

.

by Prop.

By the above we have

condition

Crs

IGws

V c aU

Similarly,

(stating

some

we have

0.4.1.

conditions

that

0.3,

K n~

,

compressed E.g.

were

K c~

and w i d e l y

Gws c~

introduced

an e l e m e n t a r y

and

K wd

denote

distributed

= {~eGws

in EHMTI3

characterization

:

~is

the

members

compressed}

2.6. is given

for those

0.6.

139

Gws

-units

in [ H M T I ] of

Gws

2.3,

2.3

that

-units

Theorem

abstract given

which

are members for

as m e m b e r ~

0.6

below

let

set.

X ~ ~(U) ~01

~au)

such

t(X)

a

-unit).

Gs

the

be a permutation such that

[HMTI]

that there

which

is no

are members

of a

Let

~k~

b e 9nY n o n d i s c r e t e

Then

there

is an a u t o m o r p h i s m

is

not

a

Gws n ~

and

Gs

-unit

-

(hence

lemma.

,

and

U

.

Let

t{q}={foq}

-units)

XCaU

t(X)

~k~ of

e n d of

Let

that

following Let

characterization

the

says

It is s h o w n

.

t ~ Is(ff~U,

O.6.1.

0.6

Gs

be an arbitrary

Lemma

Theorem

of

let

We need

At

(noncharacterizability

~kw

0.6.

is n o t

Cs

-units

for

~k3

is no a b s t r a c t

of a given

of

, for

Gs

,

i.e.

Cs

those

Theorem

-unit,

a

there

is q u o t e d .

characterization

Cs

U

~=2

of

for

let

I=

p E ~U

every

[[~U.

Let

.

there

Then

q E ~ U (p)

f

: U >~ is

U

t c Is(~,~)

and

t{q}={q}

hypotheses

of O . 6 . 1 .

for e v e r y

qE~U~(~u(P)u~u(f~

Proof.

Let

d ~

everything

and

: xEC ) since

~d

p<~d/[ : d E A t

Then i~

r ~ Is(~,~)

= ~ At

Z

(Vd,

bCAt

Z)b~d=O

e At

Z .

Let

defined d E At

as Z

z d

be as in the

z

by

We have .

: At

Let

r

Z > [HMTI] -i

At

(At Z ~ { a , b } ) I I d

Z

be

hd d ~-i

k

Id

if

d=a

if

d=b

if

d{(a,b}

and

6.2 a n d

and

by

d#a

: d E At

[HMT]

the permutation

0.3.6

Z ) : (ii),

since

b d ~u(fop)

u{< a , b > , < b , a > }

let

f

r d << x n d

= < ORgk : k e P > ,

a d ~u(p)

Z >~

Let

Let

Then of

At

Z

For

every

a,b e

140

0.6.1.

(For

the

notation

: d E At EHMT]

Z > : kEP

0.3.6

~z(d)~), we

=
Rgk

: d e be

Z >>

Then

: if

by

the

d c At

EHMTI]

6.2

At

g ~ << h d ( k d)

P<~z(d)~

every

Z >>

Z ))

we

have

hd 6

IS(~d~ ,

by

EHMT]

0.3.6

(ii))

Z > ,~),

Rgk

= (u .

Z

Then r

: r -i< h d ( X N d )

Z )

P( R l z ( d ) ~

Is(~,~)

and

by

r -i

since

: k ~

t e

of

:

: d E At

(and

: d E At

definitions

: d E At Z}

Let

g

: :

Let we

xEC

have

: d E At

t(x)

Z )

:

=

= ~(xna)uf-l(xe(b~a))Oxn(aU~(aUb))

then

t(x)

t(x)=

: ~(x)

x

: {foq

: qCx)

and

if

.

O.6.1.)

set.

Let

-unit.

Then

there

turn

Y#U

(and

that

m#n

.

the ~

proof

{

is

a subbase

U {< m , w

(Vq E

u au(f~

> , <w,m

: iE~

= {~O] w t(X)

~ X

is

and

QED(Theorem

Let

hence

not one

=

~ {
of

is

X

: iEe

in

and

)

and

t(X) the

But ranges

t(X)

and

Then

f

IYi>1

-

wEUNY

be

such

Y~U

. Let

d f =

is

there

a permutation

is

t e

~ U ~ ( ~ U (p)

t(X)

~ { <w

U

that

by

(Vq E

let Gs

and

that

and

nondiscrete

and

O.6.1

show

~hw

such

m,nEy

Lemma

shall

since

them

any

IYI>I

{foq}

~0 ~ w

normal

by

By

We

be

> }

>

~u(P))t[q}

Let

of Let

exist

0.6.

Xc~u

Y

~UNy))

= {q} -unit.

Theorem Let

m,n,w

p ~ <m

of

[~U

X c~YU

Such

. Let that

to

~yc

(UN{m,w})IId

is

: iE~

~0 = fo(p~) n -O of wn and

while

the

other

neither

the

Ws

u

not > . E

a Then

t(X) -O nw

is

Is(~,~)

are

not

not.

0.6.)

Proposition Gws wd

Is(~,

~

: d e

then

any

disjoint

3.5(i).)

t ~ r-logor

x~a

we

Then

By

d c At

Now

t{~O} w

3.i.

= r-lg< x n d

QED(Lemma

Gws n~

for

Let

(~U~(aUb))

such

since

P< R l d ~

At

U

g ~

: k ~

Therefore

of

Now

r -I C i s ( P ( ~ z ( d )

: U{hd(Xnd)

be

> .

EHMTI]

arbitrary.

~

EHMTI]

have

= r -i g r ( x )

x

see

(iii),

by

again

~

-units

0.7 have

below any

says

abstract

that

characterizations

-units as m e m b e r s

nor of

the Cs

-s.

0.7.

141

Moreover, are

there

is a

Ws - u n i t s

even if ~

~>2

and

Thus

.

By the

i.e.

even

~,X

and

Let

~(tX)

GwsC~

6%

such

Gws w d -units)

4J~ is fixed,

and

c Ws

(or

C s ~ eg

h a v e no a b s t r a c t

relative t

~ Gws n ~

Ws

that those members

Let

then

h a v e no a b s t r a c t

characterization

(noneharacterizability

and

x>l

Then

there

XEA

such that

Proof.

Let

~>~

Let

X d e

~=

~(~){X}.

and

by

X E Z d A, ,~x~)

and

not

,

x>l

Let 2.5.25

every

Ws

-unit

i<j<3

(~Vn)

there

bij(xi)=x j g'Elsm(~Vo,

and

xlUx 2

let

~d

~(VIUV2))

,

~

Then

is

s .

~,~CMn

Since

there

hoh c Id that

.

Then

- X = ~ x ~ ~x(O) statement. d U = x

Let

~]~d ~ ( ~ e U ) { x o , X l O X 2 } Gws n ~

-unit.

For every

bijEIs([~Vi, EHMTI3

For Xl d

"

Now

xO

We show that

n<3

Ux2 )

(~,

is

Xo d ~{O,1}(O),

~f(ViUV2)){x

such that

-unit

zn-x > : zEA > E Is

x>2

and

and

Ws

and

is finite.

and

assume

EHMT3 O . 3 . 6 ( i i )

a

~

I = ~x

the p r o o f of the first

t ~ Is(~,6%)

Let

then

rI(_X)~.:(~ "

V~ = ?~

is not a

.

~.

and

Let

Observing

Let

~ K

for any set

LHMT3 0.3.6,

is a b a s e - i s o m o r p h i s m By

~d

d ~U(5)

n

E

either.

f = ( < znX

statement

~t(X)

in

x>2

~X

Gws norm ,

-units)

If

which

let

Then

but

Is(4.~/d~)

-unit.

k(X)=-X

V

and

E

h< 1,0 ) = ( 0 , i >

completes

for some {x n}

t

E K

Ws

-unit

Ws

By

x2 d a { 2 , O } ( 2 )

t(Xo)=Xiux 2 d -- ~

is a

and

define

of

-unit.

since

with

the s e c o n d

=d ~{1,2}([), is a

X

~X

~ d ( s : iE~ >

• ]r~. C l e a r l y

-unit

n<3

Ws

and

~

k d= f - l o h o f e i s ( ~ , ~ )

To p r o v e

,

d~ = rlx~':6~ and

~x~)

Ws

Cs

Gws n~

f ( X ) = < ~,O > .

h E Is(~x~,

is not a

a

Then

{~ Z ~

'~E

is not a

is

(O)

FHMT3

are

t(X)

t(X)

Gws ~d,

K c {Ws

0.7.

and

Indeed,

be as in P r o p . O . 7 .

Proposition

regular

~J~ .

A

characterization

c K c Gws n ~

K -units

.

to

of

let

~n

d

For e v e r y [~Vj)

such that

6.2 we then get

g' ( X o ) = X i u x 2

.

H e n c e we o b t a i n

142

0.8.

gEIs(~o,~) we

show

Prop.

that

4.7.

element

Q, 2/~

G

(ii)

since by

Then

}

By

h(Xo)=<Xo,O

>

>

then

tEIs(~,~)

and

All

following. is a

then a

X

Ws

Cs

a

Ws

EHMTI3

.

by

Also, Let

Xo,O > ,

such

Let

Condi-

5.3.

~oX~){<

> )=( O , x l U x 2 )

that

f { <(g-lz,gy algebra

and

>

:

then

f( ) :

t ~ rlQ-lOh-1ofohorlQ

Now

.

tEIs(~,~) X=I

iff

-unit.

Assume for i<j<e ,

is not

some .

0.8.

By T h e o r e m

XCA

~=2 a

.

is a ,

because If

Ws

Gws n ~ Thus

-unit.

.

XEA

and

A(X)=~ Then

~=O be

Then

A(t(X))=a

Ibase(X) I:l

then If

and

-unit.

then

of the

x=l

base(f]L)=2

tCIs(~,~)

hence

Remark

and

uE2

By

in P r o p . O . 7

t(X)

Let

t(X)

;

t(X)

which

x~dij

for

a contradiction.

description

kinds

~

(i).

Ag=~

heIs(~,~)

are n e e d e d

,

iff

i,jEe

seven

is

by

every

is r e g u l a r

by

and by g e n e r a l

Let

conditions

base(X)#~

types

Let

O,xlUx 2 > .

f((Xo,O

~

is r e g u l a r

of

(actually,

condition

is s i m p l e

there

By

every

guished

6.2

gEIS(~o,s

-unit

for e v e r y

complete

~#J~)

By

~E

is not

implies

since

G

that

of D e f . 4 . 5

Therefore

Now

the c o n d i t i o n s

to c h e c k

satisfies .

of

h(xiUx2)=<

Subb(t(X))={{u},2} dij

satisfy

: n<3}

O.7.)

Let

-unit.

g

t(Xo)=XloX2

the

Ws

G

UGCQ

f6Is(~,~)

QED(Proposition

Remark:

since

[HMTI~

> .

Q { u{V n

sense

R = Sg{xo,xlUx2}

fCIs(~OXs215 = (Xo,O

that

element

and

CBo•

in the

and

4.7(II)

Let

It is e a s y

Q-wsmall

every

.

G ~ {Xo,Xlux2 }

A(U)Q=O

is

rlQ~

X

and

is s a t i s f i e d

rlQEIS6~

:

g(Xo)=XluX2

see D e f . l . 2 )

4.7(I)

~

that

Clearly,

of

G C S m ~L , tion

such

, Prop. O.7

of a b s t r a c t

of u n i t s

of u n i t s

0.6

defined

were

and by

EHMTI3

characterizability

in EHMTI3

introduced:

Cs

1.1. ,

2.{

,

a

of the d i s t i n -

Recall Gs

we h a v e

Ws

that ,

in

G w s c~

EHMTI3 ,

143

0.8.

G w s wd,

Gws n~

abstract By

2.1 e x a c t l y

for w h i c h Gws

(Vi,jC~)s -units

abstract under

-units.

i

of units

Moreover,

in w h i c h

of

of t h e s e they

are characterized for

Cs

, Gs

0.7

shows

this

for

Ws

,

Cs - u n i t s

property

not

0.6,

xEC~{~}

there

Ws

members

full

of

x

S~x's~x:x] s~x,s~x=x3 l

Cs Ws

the

taking

an a b s t r a c t ~

-unit

iff

[(~yEAt

since

2.~

be a n y

x

none

of t h e

even

remaining

for

(Theorem

-units.)

stronger

to a

negative

of u n i t s .

nondiscrete

Gs - u n i t

Gws

Cs - u n i t s .

-unit which Next

characterization

Zd C ) x ~ y Gws

Cs

a n d Prop.

5 kinds

Cs

by

if the

Gws norm

x

full

is a

is p r e s e r v e d

an e v e n

for a n ~

Let

[HMTI3

and

have

same holds

-units

characterizatio~

Gwsn~

Gws wd

~

Gws

is fixed.

of t h e r e m a i n i n g

Cs

do have

and

-units

-units.

is e v e n d e s t r o y e d

its m e m b e r s

-s.

since by and

full

x

be characterized

is a n a u t o m o r p h i s m

-units

is a

Gs

Gws

has

characterizations.

of u n i t s

, G w s cOmpa

of

one

characterization

this

of

abstract

as

by any one

hence

We call

exactly

-s are

a n d Prop. O.7

Gws cOmp,

the

in a n y

Gs - u n i t ;

show that

Then

and

shared

By Theorem

is n o t a

0.6

cannot

this

Cs

is an a b s t r a c t

classes

shows

of

s~x-s~x=x ] l

of

of u n i t s

is t h e c l a s s

x

Cs -s.

any kind

0.6

The

This

By Theorem

Thus

they

this one

the property

any one

automorphisms.

x . s x=x. i

has

7 kinds

those members

as m e m b e r s

because

these

and

j

3

isomorphisms.

6 types

Of

characterization,

[HMTI3

of

Gws

and and

-unit

we

as

let

xEC

.

(Vi,jE~) iff

At

Zd C = { ~ b a s e ( ~ ) (q)" : qEii}.--

A

VZ(

(Vi,j6~) Define

the

set of f o r m u l a s ~(x)

{ { (3y)[

By the -unit Ws

above, iff

-units

A ciy=y i<~

for any ~ b ~[a]

in f u l l

full .

This

Cs -s.

A c.z=z i i<~

Cs

~

,

~

(z=OVz~y))

an element

is an a b s t r a c t However,

this

A x~y]

aEC

A

is a

characterization

characterization

Ws of

is n o t

-

144

0. 9.

elementary

since

formulas.

Since

0.6

does

not

Cs

- and

set

Ws~

_c G ~. c O m P n ~ ,~ w d n ~ ~ ~ n o r m

extend

Gs

Proposition

the

to a n y

-units)

0.9.

(i)

ICrs

(ii)

ICrs reg

~(x)

in

Let

of

formulas

the

its

contains we

remaining

full

infinitary

have

5 kinds

that

of

Theorem

units

(from

strength.

~2

is a v a r i e t y ,

is

of

i.e.

ICrs

a quasiequational

= HSPCrs

class,

. icrsreg

i.e.

:

= SPUpcrsreg (iii)

ICrs reg

(iv)

ICrs

The

Lemma

# I Crs reg

detailed

outline

proof

of proof

O.i0.

Proof. is

i E

(Ax)

Let

Therefore i

appeared

~ESCr

either Assume

implies

ci(x

9 y)

ci(x ~ y).ci(-(x (C3)

and

Case x b

2

(cix

shows

To

.

Therefore

~

we

y))

,

~

y))

~

proof.

(Ax) @

ciY'-y~O

9

(Ay).

since

. .

z ~ x'ciY

x=cix

~ y)

the

,

and

(x ~ y ) - c i ( - ( x

ci(x

@ y).-(x

y)

A brief

i E AyNAx

i.e.

by

omit

A(x @

d z : x'ciy'-y

since

= ci((x

,

EAN6].

(-x) " ( c i Y ' - y ) ~ O

O ~ z ~

,

space

suppose

ciy>y

Let

~ ci(-(x

preprint

. Then

We may

or

~ x ~ y

save

x,yEA

(Ay)

-x.ciY--y

: cix

@ ciY

that

EN].

# Crs reg

by

(C3).

~ y))

= x @ y = ciO

=

This

implies

= O

by

(Co),

(C l )

Assume

@ ciY

in

x'ciY'-y~O

~ y))

Crs

in t h e

and

.

iff

found

cix=x

: ci(cix.Y ) : ci(x,y) ~ x ~ y

~>2

.

be

x'ciY.-y~O

z ~ x'-y

a>3

Let

~

Then

if

iff

can

Let

symmetric.

Case

r HCrs reg

@ ciY

~ ci(x

iEA(x

# O

.

~ ci(x @ @ y)).

9 y)

Let y)

by

Therefore

d z : -x.ciY.-y x=cix

and

. by

0 # z N ci(x

z ~ -x-ciY Bo ~ @ y).-(x @

y)

145

i. 1.

QED(Lemma

O.10.)

We note by

that L e m m a O.10

"normal

(Cl),

Bo

".

By the above

(C3)

the c o n c l u s i o n

conditions

can be dropped.

~

~

{ciO=O

conclusion

above b e c o m e s

Actually:

l

l

of L e m m a

holds.

in any

: i~}

x

i.i.

is H - r e g u l a r

is r e g u l a r

Let in

Let i~

in

~ECrs .

V V

V

be a

regular

in

~

regular

in

~.

is regular"

and

V

EHMTI3 [HMTI~

of not n e c e s s a r i l y subalgebras. They will

Definition

is

Instead

of

H-regular

"x

about 4.1,

4.1,

the c o n d i t i o n

xCV

with

and

be f r e q u e n t l y

Gws 4.2

in

eJ~ iff it

is

the

Hc~

l-regular

element

. Then

V"

in

V3.

is H - r e g u l a r of

element

in

A

of

in

is HA

is

we shall

say

"x

from context.

agrees

-s is:

and section

4.7 and

EHMTI3

elements

arises:

which

do g e n e r a t e

4.9 c o n c e r n

xEA

l.l(ix). generate

4 here deal w i t h

elements

in c o n s t r u c t i n g

Then

with

do regular

the q u e s t i o n

4.6,

~ECA

x

iff e v e r y

dimensional

used

or

is r e g u l a r

4.2

finite

Let

Let

iff every

of r e g u l a r i t y

Theorems l.3,

1.2.

~=O

is u n d e r s t o o d

question

By

~

two

set al ebras

-unit.

~, is regular

A natural

question.

[either

x

definition

ones?

None of these

~ kEx3.

The above

regular

Crs

to be H - r e g u l a r

when

satisfying

iff

iff

Then

~j~ is said

$Cr

0.10.

(~qCx)(VkEV)E(HUAx)lq~k x

Bo

Bo

is e q u i v a l e n t

1. Re ular c lindric

Definition

if we replace

in every

of Lemma O.10

(c.x=x ~ c.-x=-x)

'

proof,

false

this

collections regular

this question.

regular

algebras.

is said to be

small

in

146

1.3.

iff for every infinite

K ~ A~k

we have

(~r -~c ~) (dO --~C K) e(o )c(F)x = O . Sm~

{xeA

Theorem

: x

1.3.

element of

is small in

Let X

Before proving T h e o r e m

Definition I ~ ~ {xeA

~

}.

6~eGws n~

is regular.

be of interest

~

be g e n e r a t e d Then

~

1.3, we shall prove

some lemmas which might

in themselves.

1.3.1.

Let

HC~

:

.

denotes

the subalgebra

and

~@CA

Then

of

~

The superscript

and

with universe will be dropped

danger of confusion.

Sometimes we shall write

instead of

Sm ~

Lemma

~

1.3.2.

or

Let

Hc~

(i)

IH

(ii)

IHADm H = {O}.

(iii)

Sm~Dm H ~ I H .

Proof.

to

(i):

and

x+y~I H

~

~H(~)

or

Sm(~)

Let

.

xeI H .

Then

whenever

{x,y}~I H .

be arbitrary.

We have to show that a

xeI H

implies

(~01~ ~NH)c(O1)C(F)X=O

(~02~ e N H ) e ~ 0 2 ) e ( Q I U F ) Y = O

.

(Vy~x)yeI H

by the d e f i n i t i o n

rc_~

that

if there is no

~eCA

follow immediately

show that

DmH~ .

.

is an ideal of

Proof of

C(A)X61 H

Assume every

is regular.

: (VF C C ~ H ) c(e) ~ c( F )x=O} - ~ ~) (BO --~

Notation:

XcSms

by

Let

of

Let

(VA~ ~)

and

IH

It remains

x,yeI H .

Let

(~Gc ( aNH)c(o)e(F) x + y ) = O _

% ~ oIU82

.

yeI H

implies

Clearly,

that

@C_weNH .

.

1.3.3.

147

c(@)C(F) (x+Y)

: c ~( @ 2 ) C (~@ l ) (C(F )x + c (F) Y)

§ c(@1)c(F)y)

= C8(@2)C(@IuF)Y

We h a v e [HMT]

seen

that

x+yCI H

.

= O

,

by

The

above

< c(@2) ~ (c a(@i) C( F) x + CA

p

proves

ci(x+y)

that

< c x + ciY-

I H 9 IleJ~,

by

2.3.7.

Proof

of

that

y=O

have

that

(ii):

y E IHnDm H

y 9 Dm H

means

that

(~@c_~H)c(@)C(F)y=O

A(C(F)Y)~H Proof

Let

and

of

@~e~H

(iii) :

x~Dm H

.

Let

x9

aid by

.

Let

Fc_~a

F { Ay~H .

But

Therefore xeSmNDm H

be

IAx~HIAw

be a r b i t r a r y .

fixed. .

Then

have

is finite.

c(@)C(F)y

c(F)Y=O

,

be a r b i t r a r y .

Then

We

to s h o w

By

yeI H

= C(F)y

i.e.

since

y=O

Then

.

IAx~Hi~

(~eC_wAx~H)c(@)C(F)X=O

@c ~H --W

,

i.e.

we

we

have

by by

seen

that

xCI H QED(Lemma

Lemma

1.3.2.)

1.3.3.

Let

~eCA

be g e n e r a t e d

by

XcSm~

ideals

and

Let

HC~

Then

.

D m ~ = Sg(~) (X~DmH)

Proof.

First

we

CA

which

is an e a s y

-s

state

a fact

about

consequence

of

generator

the d e f i n i t i o n

sets

of

of

Ii~

([HMT32.3.5). Fact(~) :

Let

I 9 Ii~

be

(Fact(R)

is t r u e

the

function

Now we X~Sm

return and

let

such

D m H ~ S g ( X ~ D m H)

be g e n e r a t e d that

BNI={O)

because

~J~/I

(b/I

: b~B >

to the

proof

H~ and

DmH~IH={O},

QED (Lemma

~6CA

and

1.3.3.)

thus

XCA

Then

.

Let

B 6 Su~

and

B C Sg (~) (XNI)

is g e n e r a t e d

{b/I

by

: b9

and

is o n e - o n e . ) of L e m m a

be fixed. XNIH~Dm H

by

Then

by L e m m a

1.3.3. iH E

Ii~

1.3.2.

D m H = S g ( X n D m H)

~J~

Let

,

,

Dm H E Su~,

Hence by

be g e n e r a t e d

by Fact(e)

D m H 9 Su~)5

.

by

148

I .3.4.

Lemma

1.3.4.

Let

~>O

Statements

(i)

b.

x

is

{i}-regular

e.

x

is

H-regular

for

some

d

x

is

H-regular

for

all

9

and

x

Proof.

is

Proof

of

exists

bERgkQRgq f~P)

is

we

in

O

(i):

If

=

d.

GCHCa x

, holds

then

. Then by Ax#O

~>0

a.

G - d.

that

finite.

the

Then

is

obvious

We .

Let is

k~P)Ex--

by

observing

. Let prove

Since

FC

that H#O

there

P ~ F~(HUAx)

finite,

we

have

Then

.

since

kex

H -regular. H#O

F -regular, not

(ii)

was but

essential while

~i

0 -regular). of

the

-regularity are

following:

-regular).

is

assumption

by

Suppose

By (AxUHUF)Ik~P) i q~P)~x

is

it

the

c qEx

P

is

the

kEx)

R -regular3

fixed.

have

element

Wlu~{l} c.

if

hold

direction

-regular

in

and

-regularity.

this

~

is

-regular.

Since

x

that

a.

be

x

. G

HUF

is

HUF

of

of

~

is

-regularity

-regularity d.

x

<~-Ax)

i -regular

fact

prove

P~w

every

of

b

that

a

and

(HUAx)Ik

(~p)IfUp•

proves

H

x

is

and

of

-regular"

~

this

Axlq~k]

to

by

that

Proof the

Let

This

(namely:

~=

x

.

G -regular.

(HUF)

kEV

qb

P~(a~Ax) Note

is

(P)~x

and HOF

=

HC- - W a

and

of

enough

that

Let

~

G

x

is

.

nonempty

-regular

suppose

H -regular.

k~P)Ev

L

H -C- W a

.

subsets

H -regular

is

and

Notation:

It

iE~

# 0

H 9

"H-regular

(VLCRCa)Ex

by

some

nonempty

iff

(ii) :

is

direction

O#HCa

be

H-regular

is

and

for

difference

a)(VH#O)(x

and

G

.

i~).

(VqEx) (~kE1~)(ERgqnRgk H

xEA

equivalent.

regular

symmetric

x

(in

are

is

Let

that

below

x

(ii)

The

a.-e.

a.

e.

(VF c

and

~O~EGws

,

of

equivalent

.

e.

~

d.

holds

then

d.

~

e.

can

be

by

the

seen

present x

lemma

implies

since

a.

definition by

choosing

~

and

by

H b. of

=

c. H -

H = Ax

.

i .3 . 5.

149

Suppose

Ax=0

.

Let

qEx

existence

of an e l e m e n t

~=~ kEx

The

QED(Lemma

1.3.4.)

above

Lemma

1.3.5.

(i)

H -regular

(ii)

If ones

in

~H

xCA

is

H -regular

Let

x,y6Dm H

is

enough

x

and

y

that

such

kE(xny). if

both

~EGws

Suppose

then be

to s h o w

let

-x

is

H -regular. .

Then

Let

by L e m m a

.

by

Since

xny

is dij

GDH

both

x

is

b

Further,

(by

qbEV

,

Lemma

. H -regular

and

V

kEcix

(i)

and

k EV

Therefore

cix

@

GDH#O

and

is

it Both

kEV

be

we h a v e Obviously,

HC_~ ,

.

GlkCq6cix

k~Ex

H

is proved.

i,j6~ .

Then

RgkNsgq#O

because

Then

G

G -regular,

for e v e r y kEV

Now

and by

qE(xny) are

is normal. .

.

/JSECrs~

H - regularity.

By this,

Let

if an e l e m e n t

by

G -regularity

G -regular.

1.3.5.)

1.3.6.

( V H ~ a ) D m=~- X

hence

.

of

y

H-regular,

some

QED(Lemma

~=~

in ~

in any

(ii)

Let

and

G -regular.

too,

1.3.4

.

for

and

k~Ex

generate

G d HOAxOAy

instead

G1kbC_qbEx

,

ones

By d e f i n i t i o n ,

H -regular

iE~

x

and

the

be n o n e m p t y .

elements

O*HC~_

and

of

k~EV

-regular

H -regular

is n o r m a l

H#0),

HEn H

G - regularity

then ~

,

implies

e. ~=~ d..

let

G -regular

Thus

q~Ex

generate

and

x,yEDm H

G1qC_k

Rgq~Rgk#O

Then

that

and

then

~J~CGws

are

.

"

Let

by

that

elements

is n o r m a l

kEV

such

~EGws

Proof.

is f i n i t e

b

prove

Let

~

and

Let

~EGws n~

Sg(2~) (XADmH)

be g e n e r a t e d Then

by

XCA

{][ is r e g u l a r

Suppose

if e v e r y

element

of

is r e g u l a r .

Proof. H=IUAy Every

Assume

the h y p o t h e s e s .

. Clearly element

of

yEDm H XNDm H

.

Let

Then is

y6Sg(~)X

be a r b i t r a r y .

yESg(~) (XADmH) ,

H -regular,

by L e m m a

by our 1.3.4

Let

assumption. and by

H#0.

150

1.4.

Then

y

is

that

y

is r e g u l a r .

QED(Lemma

H -regular

(Theorem

Let H -atom

of

~

follows

H=IOAy

,

this m e a n s

from Lemmas

1.3.3,

1.3.6.

xEA

.

Recall

if it is an a t o m of of

~

the

from

[HMT3

Boolean

that

algebra

x

is an

~6H ~

of

.

1.4.

Every

normal every

Let

Gs

a.-b.

a.

If

b.

Let

H

Since

generated and

let

by

by

its a t o m s

its a t o m s

is r e g u l a r .

(There-

is r e g u l a r . )

HC~_ .

Suppose

~J~

is r e g u l a r .

A=Sg

At C I H ~

-

hold.

is f i n i t e

then

YC_At ClH~]~

be a set of r e g u l a r

elements.

Then

is regular.

A = Clog

it is e n o u g h

it is e n o u g h

generated

below

~ ) Y

Proof.

Gws

~ E G w s n~

Then

Claim

immediately

and

elements

fore

Hence

Since

1.3.)

, ~ECA

Corollary

(ii)

1.3

HC~

H - closed

(i)

1.3.5.

1.3.6.)

Now Theorem QED

by L e m m a

we

to p r o v e

to p r o v e

the

have

that

(ii).

Let

following

(i)

is a speci~al

~eGws n~

case

of

By T h e o r e m

(ii). 1.3

claim.

1.4.1.

(i)

At C l H ~ ! Sm~

(ii)

Let

Hc a

and

for

every

H!~

~eGws nOrm

and Then

~ECA every

. H-atom

of

~

is

regular. Proof.

Proof

infinite 1.iO.3(i).

and

of(i): let

Fc_~

]aN(HOF) r ~

Let

y Then

and

be an

H -atom

C(F)y gy#O

is a by

IK1~

of

4~ .

HOF

-atom

.

Then

Let

KCAy

by

[HMT3

by

[HMT3

be

151

i. 5. 8 Y~Codol

i.iO.5(i),

which

8 C(F)Y~Codol

implies

Thus

Then

E CI(HuF)~

which

implies

We h a v e

that

y E S m t~ . (Note

> cic(F)y

C(F)YCAt

CI(HuF)[~

1.10.3,

1.iO.5

is true

because

hold

for

seen

H -atoms

in t h e i r

proofs

.

where

Let

therefore

A ( C ( F ) y ) = ~N(HOF) C(F)y

KnA(C(F)Y)~O

and

H

i E KnA(c(F)y ) c~c(F)Y=O that

is i n f i n i t e ,

no c o n d i t i o n

,

implying

by

CHMT]

too.

This

IHl<w

is

used.) Proof ~A~

(ii)__i:

If

e<w

Let

HC ~

.

Let

y

Ay=~H

;

and

if

Ay=O

.

else

of

and by

~

every

.

cofinite

y ~ Codol subbases

,

QED(Claim

Remark .

Let CoX

Namely,

Gws

Prop.

of

(3) T h e

there and

Clearly finite.

~

~

a

~ w b

with Let

is an

Gs

V

1.4

If A y = O

with

or

1.10.5

(i)

,

since

then

by

in o n e - e l e m e n t

1.3.6,

(This

last

and

the

~

that

in

1.4.1

the

suppose

V

V

but

(ii)

is true

(See a l s o Let

~

.

elements

that

~NH

H -atoms.

b

condition

.

V ~ ~1u~2 (~)

: (~--H)IfCO}

. Now

in

following.

regular

if we

(~)

and

the

V ~ ~2u~{2}

A(~2)=O

1.4

from

(ii)b

unit

is O - r e g u l a r

non-regular

b ~ {fE~ of

IHI<~

if we o m i t

, since

is not

(even

H -atom

x

follows

some

Ay=O

[HMT]

is normal.

false

full

~ ~ ~x{~}

be o m i t t e d Cs

becomes

(ii),

in Cor.

~

Suppose

in g e n e r a l . )

in

el E A t ~]Z

by

contained

since

Clearly,

This ,

are

by

1.17.

1.4.)

be the

1.3.5

condition

exists 9

1.3,

y

Gws

O -regular

Then

cannot

is r e g u l a r .

1.3.5

: f(O)=O}.

Let

~T/~&(~@V).

element

EHMTI] either

,

is r e g u l a r

for

~

in g e n e r a l .

4.2.)

regular

let

Then

y

of

by

y ~ codol

regularity,

Lemma

is n o t

(2) N o n e

then

elements

true

are done

H -atom

then

and C o r o l l a r y

(i)

x = {fE~2

for

the

implies

1.4.1

= ~2

Ay=~--H

is not

1.5.

be an

dimensional

all

which

implication

H#O

If

t h e n we

Y

iff

be

is i n f i n i t e ) :

Namely,

where

is r e g u l a r

of

let

a~e

O = ~x{O} H

is

.

152

1.6.

Remark

1.6.

Definition

(Notions 1.6.1.

of r e g u l a r i t y )

Let

KCCrs

. Then

K Oreg

{ { ~EK

:

~

K zdreg

~ { ~6K

:

(VaCA~Zd

K ireg

~ {{~EK

:

(~aEA)(~iE~)

a is

{i}-regular}.

K creg

~ { ~EK

:

(VaEA) (~iee)

a is

{i}-regular}.

It s e e m s

that

is O - r e g u l a r } .

among

interesting

ones

are

Propo@ition

1.6.2.

the

Let

Gws reg = Gws zdreg

(ii)

GwsCOmp

(iii)

I Crs

(iv)

reg

Let

~2

follows

from

1.3.4(i).

that

such

f#k

We

Then

and

that

-regular,

Proof

x

fO=kO f6W

we m a y

assume

Then

A(d,x) lh~fEd,x

~Gws

zdreg

Let

~ECrs creg

-regular Gs ~

by

by

_~ Gs reg. ~

(i):

Gws creg

Gws ~

D

D I Crs ~

and

let

Gws reg = Gws Ireg

~Gws

zdreg~

By

such

that

kEW. fcW.

~Gws

Since Let

Crs c r e g

is

and

and

,-do1}

is not

c Crs zdreg A.

Thus

is O - r e g u l a r

~ d= e •

Ax=0

Let

d : ex{l},

fEx

WCSubu(~),

x 6 A~Zd x

there

and

by

h~x

x

is not

and

kE-x.

0fed.

by Ax=O. Hence

seen

Then

x

iCAx

V d= {O}U{~}

xEA

that

O-regular.

iEAx.

is

is not such

can be

~J~E

Ibase(W) IA2

-x be

h ~ k~IEl~

d,x

Let

ireg~

fO=kO,

A-x:O

dE{d

w e have

Then

that

there

that

= Gws creg

Gws z d r e g c Gws ireg.

{i}-regular.

f,kEl ~

proves

creg.

Let

:

show

~EGws~

and

.

N o w we

or

A(d,x)#0.

~ECrs

of

are

either

By

zdregularity.

= $PCrs ~

we h a v e

A(d,x)=2.

the m o s t

oreg

is not

there

Then

of r e g u l a r i t y

D Crs creg

show

(Vie~)

l-regular. By

.

and

= Gws ireg

= I Crs z d r e g

Proof.

such

~A2

notions

D I Crs reg D I Crs c r e g

D Crs zdreg

E G w s ~ 3 w s ireg.~

above

= GwsCOmp

D I Crs •

I Crs c r e g

a is O - r e g u l a r } .

cregularity

(i)

Crs

A)

and

as

follows.

is {i}We ~{

show @@V.

2. I.

153

Then

9

is regular by

{0}EC

is not

Proof of xCA, ~Gws x

(ii) :

Let

and suppose

and

imply

hEx

kex.

by

~]~CGwsCOmp reg AxlkCfEx._ Ax3kCf.

We show

Let

x

is 0-regular.

Then

h c i~

c o i n c i d e s w i t h r e g u l a r i t y in ,[AGN23

and r e g u l a r i t y of

(ill)and

(iv).

as defined

Cs N L f .

In

in [ A G N I 3 , [ A N I 3 , [ A G N 2 3

Gws~ALf~

c o i n c i d e s w i t h regularity.

In

"i-finiteness"

Crs nLf

"i-

of [AGNI3 c o i n c i d e s with O-regularity.

2. R e l a t i v i z a t i o n

D e f i n i t i o n 2.1.

Let

~%ECrs

(i)

~d A d rl Z = rl Z = rlA(z)

(ii)

Riz~d

.

Let

d = (xnZ

Zcl ~

and let

~:

[@Z.

: xcA > .

R1 Z A d RI(Z){)~d Sg(~)

We shall omit the s u p e r s c i p t s

4)5

rlz: A

and

A

and

if there is no danger of

confusion.

The above d e f i n i t i o n of f r e q u e n t l y use the fact that Z C Zd Sb i~.

by

we have the following c o n n e c t i o n s with earlier

The n o t i o n of "i-finiteness"

~niteness"

Let

1.6.2.)

By P r o p . l . 6 . 2

JAN13

h d fko'O

lOAx~kChEx

We have seen that

QED(Proposition

of

since

~CGws ~

Now

To save space, we omit the proofs of

papers.

~ ~ Gws ~

0-regular.

f,kel~ , c~

( V f , k e l X ) [ f O : k O ~ f=k3.

r l z " agrees with A (~,~Z ~) rlzCHO

[HMTI36.1. for any

We shall

~]~6Crs

This fact follows by the proof of [HMT32.3.26,

and a

d e t a i l e d proof can be found in IN1]. P r o p . 2 . 2 below says that r e g u l a r i t y can be d e s t r o y e d by

rlz,

154

2.2.

unless both

A[V3z=o

and

needed by P r o p o s i t i o n

Proposition

2.2(ii)

Let

(ii)

For every

6~ECrs reg

~

Proof of

and

Let

pEy

and

by

y

Then

~ C C s reg

~Z ~

and

y

there are an ~Z~)~

~ E C r s reg

be arbitrary. by

is regular.

ZEA

such that

is regular

in

~,

in

Cs[ eg

and

Z E Zd#]~

by

y

is regular

By

~)~, and therefore

too. This proves

Let

that

~

iE~.

ciY~ = (c~iy)NZ =

(iUA(~)y)Ip~q. A~y = A~y.

Let

Z E A.

and therefore

be such that

is regular

yCA

We show that

by

~6

is not regular.

and let

Then

Z E Zd{%

(iUA(~)y)Ip~q

in ~ .

Then yEA

~

q E i~

and

qEy.

Let

~ E C r s reg

We have seen

is regular,

since

was chosen arbitrarily. Proof of

Let

~

(ii):

be the

{X,Z}_c Sm @5 X

Let

q E i~ = Z

Z!l ~ and

that

~2

A(~)y = A(~)y.

we have that

y

(i) :

yER

Thus

Z E Zd6%.

such that

ciY~ _< c~Z = Z

= ciY.~

are

(iii).

there are

Z ~ Zd Sb 1~

Then

Both of these conditions

~ ICs reg.

For every

{~Z~)~ .

and

and let

~w

~Z~]~c Cs

Proof.

hold.

2.2.

(i)

(iii)

ZCA

since

Let

Cs

Since

X

and 1.3.

Clearly

X#O

and

Let

(VFc_~) (~i~F)

regular by Theorem

= Z = ~2.

.

p ~ ~xl,

with base 3 and generated

since

XcZ.

~

XeR

X#I ~

Z

c~c(F)Z=O,

are regular by Let and by

~ ~ . ~X:O ~w

X ~ ~2 (p) by

{X,Z}.

Then

~ECs

Therefore since

QED(Proposition

2.2.)

{]L is 1~ =

(despite the fact that A~X = a). we have that

~

is not regular,

by EHMTI34.3. (iii) of P r o p o s i t i o n

Z ~ ~2.

and the same holds for

AX=AZ=~. Then

and

2.2 is a consequence

of Prop.4.11.

2.3

.

155

About

Proposition

Proposition Proof. ~d

2.3.

Let

> }.

A = {V,O,{(O,1 _

x d

[HMTI]2.10.

), < 1,2 >, < 2,3 > } show

> }, {< 1,2 > ,< 2,3 > ]}.

Let

> }EN

{< 1,2 ) } E R I v ~

~J~ECrs 2.

and

We

{<0,1

Then

d Y.Cl(dol,CoX)EN"

~{

see

C r s 2 # R~Cs 2.

V d {(O,1

[~(~){<0,i

Then

2.3 b e l o w

that ~ R s ~ECs 2

y = {( 1,2 >,( 2,3 > }EN.

z = {< 1,2 > }

for a r b i t r a r y

since

~[eCs 2

if

2. and

d

and

But

~ d ~V@~24.

We h a v e

suppose

~ E C s 2.

AC_N.

d

Then

A_CRIv~ .

Let

z = Hence

Then

z~i

shows

R~ cs 2

Q_ED(Prqposition

2.3.)

3.
About (iii) says

Propositions3.4-3.5

says that

eralize

that under

to

Gs

of the b a s e s implies

[HMTI33.6

that

some .

be

Prop.3.5(i) finite

Lf

cannot

cannot

1. Let

be a f u n c t i o n .

Note 2. Let (i)

that

and

~(eU)

~,~ECrs

.

not

F

some

is a s t r o n g

Gs -s,

but

does

the c o n d i t i o n

from

with

Prop.3.5

in

gen-

that

[HMTI]3.6,

Dc

Prop.3.4

and

one 3.5(iv)

[HMTI]3.6.

we d e f i n e : x

is a set of

for a n y

F E IS(~,~)

one-one

to

[HMTI~3.6

that

be r e m o v e d

RgqC_Dof}

Let

generalize

implies

Then

c e(Rgf)

[HMTI]3.11(1)-(3).

hypotheses

F is a b a s e - i s o m o r p h i s m for

(ii)

: qEx

see

be r e p l a c e d

3.1.

~{foq

does

additional

Definition f

below

if

sets with

U

functions and

> .

~.

DoF=A.

(F = A I ~

and

base(~)

_c Dof)

f.

ext-isomorphism

if

F = rlA(~u)

for

some

U c base(~). (iii)

F

is an e x t - i s o m o r n h i s m

if

E = rlA(v)

for

some

V c {~

156

3.2.

(iv)

F

is a

(strong)

(strong) (v)

sub

ext-isomorphism

is t h e d u a l

-isomorphism (vi)

F

is a l o w e r

above

3.2.

(i)

Let

f

and

~'~ECrs

h

Proof.

is n o t

(i)

Proof X =

of

.

and

Clearly

z = [qE~W

Then

~

= W~z

and

-isomorphism. = {koq

: qCx}.

F

is a

(strong)

strong

with

(strong)

sub

F = k-l~hot

for

ext-isomorphisms

let

and

some basek,t.

[HMTI]3.15.

~]~ECrs

, Ws

is a s t r o n g

(-base)-

ext(-base)-isomorphism.

EHMTI]3.5

Gs

is

.

Assume

, C r s ~ eg

~EIs

Gws n~

G w s wd,

ext-base-isomorphism,

~]~ecs[egnLf

and

and

h c Is(~,~)

such

that

f r o m the d e f i n i t i o n s .

~>O.

Let

@~s

by

-x.

~

IAI

E Is(~,~)

: qoE~} be

-x

there

Then

~

some

Let

k

(VuCU~)

f*~

~eW

(~+~)

Cs

that

~.

there s

is

h E

conditions and

~ W ~ U Let

Since : W~

by

~ U

~EcsregNLf

is

such of

+

satisfy

IUI,

there

U d

show that

B : Sg{z}.

and is

for

we

and

~ E Is(~,~). h(x)=

l~u~l.

and

~ ~ <

[HMTI]3.18(i)c)

f e WW

Then

~ d First

is a b a s e - a u t o m o r p h i s m

E Is(~,~)

is,

KE{Gws

h(x):

rl(~W)

we have

h.

and

because

IWI = x

Let

base-isomorphism

function

AI~

~

that

.

: ~.

Let

Let

= ~<w

IW~•

some

agrees

easily

[HMTI]3.18(i)c) Then

and

some

a base-isomorphism.

(ii):

such

for

if

if

~'~6% E K.

: qo<~}.

E IS(~,~)

is a

and

There

follows

of

d {qE~U

h

implies a>O.

that

F = goh

base-isomorphism

Then

205EK Let

-i

be a o n e - o n e

GwsC~

(ii)

F

definition

Lemma

g

o f ext,

if

-isomorphism

The

ext-base-isomorphism

such

we have

fof c Id.

EHMTI]3.1.

By

~(z)

h = rlA(~w)-lo~orl(eW).

Then

h o h c Id.

Assume

: U >~

such

U

(~qEx) ( k o q ) o : U .

that

that Then

that

z ~ xn~W.

~+~ ~ W and

~ :

h

= h E

is a b a s e -

-x : h(x)

=

3.3.

157

This c o n t r a d i c t s

( V u E U ~ ) (~i<~)k(i)=u.

Thus

~
h

is not a

base-isomorphism. Q E D ( L e m m a 3.2.)

By Lemma 3.2(ii)

it is m e a n i n g f u l

to ask w h i c h

isomorphisms of

b a s e - i s o m o r p h i c a l g e b r a s are actually b a s e - i s o m o r p h i s m s F r o m the proof of [HMTI33.6 csregnLf <~N~.

themselves.

it follows that every i s o m o r p h i s m b e t w e e n

-s is a b a s e - i s o m o r p h i s m ,

if one of them has base of power

P r o p o s i t i o n 3.4 b e l o w g e n e r a l i z e s

this form of [HMTI33.6 to

Gs -s.

D e f i n i t i o n 3.3. (i)

Let

~ECA

teristic ~

(ii)

1

has

Let

.

6~is said to be of r e s i d u a l l y nonzero charac-

iff a

nonzero

~ECrs

strongly

~Ec_

.

Piei~ i

for some

~IcA~

such that each

characteristic.

We say that

ext-isomorphic

to

Note that if every subbase of

6~

is b a s e - m i n i m a l

any

o~

Crs

except

has power

if ~

is not

itself.

<~A~

then

~4 is of

r e s i d u a l l y nonzero c h a r a c t e r i s t i c .

P r o p o s i t i o n 3.4. ~Gsreg~Lf (~)

~

Assume between

(3)

Gs -s) Let

~, ~ E

be of r e s i d u a l l y nonzero c h a r a c t e r i s t i c .

A s s u m e that either Then

(2)

( g e n e r a l i z a t i o n of [HMTI]3.6 to

~

is f i n i t e l y g e n e r a t e d or

is strongly e x t - i s o m o r p h i c ~

and ~

and

A s s u m e that either

~

Gs

.

Then every i s o m o r p h i s m

is a b a s e - i s o m o r p h i s m . ~

is f i n i t e l y g e n e r a t e d or

Then every i s o m o r p h i s m b e t w e e n -isomorphism.

is atomic.

to some b a s e - m i n i m a l

are b a s e - m i n i m a l . ~

Zd~

~

and

~

Zd6%

is atomic.

is a lower base-

~58

3.4.1.

To p r o v e

3.4 we

Lemma

3.4.1.

ually

nonzero

is f i n i t e

Let

~

U

such

that

Subb(~) be If

let

x+ = d {mlq

: qEX}.

define

U{N r

: rEmn}.

then

~ O.

Then

iEI some

with jEI.

regularity A = SgX. the

so

,

qE~U i

of

x

Let

~

requirements

Q E D (L e m m a

Let

(~iCI)k i let

G(x)

Zd~

Clearly

9

remn

and

k I~

{ {x}

it f o l l o w s

we get

qCx.

~(~)U{Gx of L e m m a

if

Then

and

: icI}}.

We

show

also.

i=j

We h a v e

seen

k E

xEA

be

hence

~ {Yv

fixed.

06~. For

Let every

: v
that

show

:

iEm]

x =

q E Z r , N r"

Thus qeN r

we get and

t there

mlq so

= k .3o r

m l q E x +.

xESg(~)G(x).

Clearly,

~ i ~.

r d k ?i l o ( m l q ) E m n "

Since

qEZ r

: A xEA.

~{

Let

d ~{ O r : siYi

Set

that

3.4.1.1.

G(x)

Z

: x~X}.

Let

: k i orCx +} "

set

q~Zr

G

every

(~gCG)A(~)g~l.

qE~U i .

is fin-

x ~ {0,i~},

we

By

for

Let

~>O,

Let

q C Z r . N r. +.

eJt

n>O

Nr d u{~U i

Say

if

Then

is a f u n c t i o n

: n > ~ - U i.

rEmn

n>O.

(~icI) IUiI=n.

m>O.

Set

every

qEx.

qEN r

m#O

3.4.1.1.)

of r e s i d -

lemmas.

xCSg(~)G x

I. T h e n

: m]qE{kior

and

Since

and

Further,

Moreover, there

: (3i~I)qo=kiv].

For

suppose

~.

and

IG(x) I
: rEmn} 9

klorEX +

two o t h e r

characteristic

some

rEmn.

Yv ~ { q E l ~

= ~{Zr'N r

is atomic.

In fact,

Suppose

Let

x c Sg(~)G(x).

Conversely,

for

m:O

Z r : {qEl ~

generated

be of c h a r a c t e r i s t i c

be of

that

Clearly,

Thus

Zd~

establish

~.

such

G(x)

lemmas.

finitely

IGxI
: I >-~

v
we

so is

~EGsreg~Lf

m ~ Ax.

Then

~EGsregnLf

Let

otherwise

that

first

then

E I(nbase(~)) Let

be

some m o n a d i c - g e n e r a t e d

generated

Proof.

following

a characteristic.

Let

for

Sb Cl NI~5

Let

has

3.4.1.1.

itely

the

~GsregmLf

3.4.1,

~]~C~EGsreg

use

characteristic.

if

To p r o v e Lemma

shall

G

and

~

is an for By

Assume satisfy

3.4 .1.2.

159

Remark: with

If in Lemma 3.4.1.1

we replace "of c h a r a c t e r i s t i c

"of residually nonzero c h a r a c t e r i s t i c "

becomes false. N a m e l y let

a~.

s u b b a s e s f i n i t e such that no (Being m o n a d i c - g e n e r a t e d Lemma 3.4.1.2. Suppose

Let

~

~ ~

Let

XCA

By EHMT]2.2.24 = {a (Y,Z)

~eCA

be such t h a t

Then

a~(Y,Z)=O

c(~)d(~•

Therefore

IC]<~

Zd~:

for every

Then

P.)

Y,Zcx_

IXl<w,n<w.

IZdO51<~.

IXl<~

Sg(~s

and

where

(VxeX)Ax~l. C =

Let the c h a r a c t e r i s t i c of

for every

by

is m o n a d i c generated.

A : Sg X,

: yoz ~ X, ~<(a+l)nw}.

Then

w i t h all

be f i n i t e l y - a n d m o n a d i c - g e n e r a t e d .

we have that

n>O.

~6GsregnLf

is not p r e s e r v e d under

fJ~ is of nonzero characteristic.

Proof.

then the c o n c l u s i o n

Then there is Crs

n>O"

~>n. since

Then

Let

~<(e+l)nw,

6~ be •

a (Y,Z)~c(•

]Zd~l = ISg(~s ~)%)Cl<w.

QED (Lemma 3.4.1.2.)

Now we turn to the proof of Lemma 3.4.1. Let the c o n s t a n t term

o

c (n)d(n• -

On if

n#O

and

of the d i s c o u r s e

n

)d((n+1)• ~CGws

then

language of Note that

" o~ = o{V

n<~nm.

We define

CA -s CA a

: Ibase(V) l=n

as

~ OO=1

and

and

vesubu(~)}.

n

Let

~ e C A a.

Then

6% is of r e s i d u a l l y nonzero c h a r a c t e r i s t i c

(~aeAN{O})(~ne(an~)Nl)

iff

ano~#O. n

Let ~ e G s r e g n L f

be finitely g e n e r a t e d and be of r e s i d u a l l y non-

zero characteristic.

Let

9 GsregnLf

~

is a

Gs

a

since

-unit and

n
n ~

~

Let

by

~ n ~ %~go(n)~g.

~CGs

a

Then

~n e

w h i c h implies that

is regular by 2.2(i)

since

o~ n

~ is regular

n

and since Then

o~ e Z d ~ . n ~

Also,

c P(/~n : ne(an~)Nl > for some

is of r e s i d u a l l y nonzero characteristic.

o~n ~ ~

Let

for some f i n i t e l y and m o n a d i c - g e n e r a t e d

The c h a r a c t e r i s t i c of 3.4.1.2.

6% ~

Therefore

~

is nonzero and therefore

IZd ~ n I<~

by

~

! ~-

Hence

~

n < an~,n#O. , by 3.4.1.1.

IZd ~ I < ~ Zd ~{n

by is atomic.

160

3.4.

We h a v e

~_c

p <~ n

: n~(~i~)--i >

by. [HMT]2.4.3.

By

{~.

: i<

1

<~w}

C A

we h a v e

Therefore atomic.

C D {( O

: ne(~)~l

(~n~(a~)~i)(Zd~

Then

QED(Lemma

Zd~

n

> (i/b)

is atomic)

is a t o m i c by

that

b~B i}

~

is

~

3.4.1.)

residually

nonzero

P r o o f of(1): is atomic.

Then

At Zd A.

Let

Zd~ Y

a#O,

either

(][,~EGsregALf

~]L is f i n i t e l y

is a t o m i c

in b o t h cases,

: T ~ Subb(~) Such a

hence

Let

be of

characteristic.

Suppose

IY(a) I
Then

i~(~w)~l,

implies

N O W w e t u r n to the p r o o f of P r o p . 3 . 4 .

and

:

Y

generated by

be such that

exists

3.4.1.

s i nce

Zd~

Let

(VaET)[ey(a)

by the f o l l o w i n g .

(3n~(eAw)Nl)ano~#O

or

Let

T

c a aCT.

~]~ is of r e s i d u a l l y

n

nonzero

characteristic.

IY(a) l=n.

Then

U ~ ORGY.

Let

rlw~HO(~,~) Then

Thus

aney(a)#O

~Y(a) ~ a W=eU.

by

since

zeZd A, by

a

We show that

W E Zd Sb i~ ~eLf

.Then

.

Let

aSz

rlWEms~ beh~{O}

a=c(Ab) (a.b),

hence

and

~ a.

Thus

We h a v e

ext-isomorphism. Suppose Let

that

aET

V~U

and

= ~(VAY(a))

n<m.

We have

#b~Y. = {~Y

hence

C(Ab)b=~y by

~U~b=O

IVny(a) I=n c V.

Zd A

is a s t r o n g

is b a s e - m i n i m a l . ~

~(~V)~.

Let

YCSubb(~) for some

Then

be of r e s i d u a l l y

and

: IY(a) I

Y(a)cV

V=U. We s h o w that

U ~ base(~)~Y.

b ~ A~{O}. .

n ~

which means

is b a s e - m i n i m a l .

since ~cGsregnLf _ req ~Gs -~Lf .

~,~EGsreg~Lf

~

Y(a)ESubb(~).

(~aET)Y(a)

Let

since

Then

nonzero

b.

Y(a)ESubb(~)

rl W

Let

Then

z d: C(Ab)

a' = e V A W A a = ~VNeY(a)

by

Then

by

seen that

Let

~ ~ ~W ~.

a E At Zd A

br]eY(a)#O

Then

Aa=O.

be arbitrary. Let

rl(~V)~Is~.

~EGsregALf

: YCSubb(~)}.

: YESubb(~)}

N o w let

s e e n that

Suppose

rl(~U){Is~, Then

O # a N o ~.n

and

Let

We s h o w that

VAY(a)ESubb(~) by

P r o o f of(2) :

Then

.

is such t h a t

and

At Zd A = {~Y

~eGs

a' ~ r l ( ~ V ) r l ( W ) a .

We have O#a'<_o~n by

Clearly,

.

for some

Then

bmW#O.

Y(a)ESubb(~),

is r e g u l a r

is atomic. ~Y(a)

for some

Then

O

At Zd A =

characteristic.

3.5.

161

Assume

that

Ys and

have

nonzero such F

and

and

let

therefore

~ a ~, % ( a ) ~ We

~

h

(3) of P r o p . 3 . 4 QED(Proposition

Proposition

below

are

The

Clearly,

[HMTI33.6

F ~ U[F a

since

above

and

(discussion

h

: At

the a b o v e .

9 At

and

there

" ~

are

a.-c.

a.

~

b.

~

c.

There

The

condition

is n o t

).

is of r e s i d u a l l y

is

Fa

Zd A}.

Zd A > ~

At

(i)

Proposition

: Y >~

W

Then

Zd B,

and

and

h=AIF.

(2).

are

relationship

in t h i s

of

other.

this

Prop.3.5(i)

duality.

the c o n d i t i o n s

finitely

lower

in P r o p o s i t i o n

nonzero

Namely: generated

" ~

base-isomorphic

Gws

3.4(3))

characteristic"

Let

~

and

is

x~. sat-

~,~E•

in P r o p . 3 . 4 ( 3 ) .

and

~,~

Moreover, ~

to ~ .

sub-base-isomorphic

is f i n i t e l y

isomorphic

~

to e a c h

end of

below.

base-isomorphic. to b o t h

3.10 a t t h e

b e of r e s i d u a l l y

two

is n o

is n e c e s s a r y

E

to b o t h

generated

or

Namely:

Let

Gs~egnLf

no

(further

Crs ~

Zd~

and

~.

is a t o m i c "

~>i

which

~

and are

not

~>O. lower

is s u b - b a s e - i s o m o r p h i c is h e r e d i t a r i l y

nondiscrete

~>i).

"lower

Zd B

Then

h 6 IS(~a~,~h(a~

: a E At

of

in P r o p . 3 . 4 ( 2 ) , ( 3 ) .

there

There

by

h(a)

is a b a s e - i s o m o r p h i s m .

condition

isfying

(iii)

Let

an asymmetry

3.5.

Then

if

by

hence

Let

3.4.)

necessary

(ii)

Then

in a k i n d o f d u a l

Proposition (i)

2.2(i).

Is(~,~).

Zd A,

WdSubb(~),

is a c o n s e q u e n c e

3.4

exhibits

h e

is b a s e - m i n i m a l

base(~)

that

Let

a c At

some

since

R1 a A 1 h -c ~a.

seen

Then

for by

characteristic.

>~

base-minimal.

a ~ ~Y.

h(a):eW

IY I< @ @ ~

: base(~)

section

are

E csregnLf

that

We have

~

base-isomorphism"

cannot

be r e p l a c e d

with

"base-iso-

182

3.5.1.

-morphism"

in P r o p . 3 . 4 . ( 3 ) .

Then

are

are

there

"Lf" For

GsregnMn~

not base-isomorphic.

between (iv)

~,~c

GsregNLf

cannot every

-s

If

there

Let

~6w~5

such that

H<5S~

be arbitrary.

~ ~ ~

then

every

but

they

isomorphism

is a b a s e - i s o m o r p h i s m .

be r e p l a c e d

e~

Namely:

by

are

"Dc"

in P r o p . 3 . 4 ( 2 ) , ( 3 ) .

isomorphic

3cs[egNDc

Namely:

-s w h i c h

are

not base-isomorphic.

Proof. Lemma

Proof

of

3.5.1.

ultrafilter

3.5(i) :

Let on

~

I.

udCIs~,

be a

Let

ud { < { q E e ( I u / F ) Then

ud:~E

is

: iEI > /F

the

strong

Proof. (F, ( U and

Let

:

In fact,

f,8,g

for any

ga = {qC~(Iu/F) so it is c l e a r

that

so

{jEI

: (ViEAa)(

of

a,

=@W.

Hence

QED(Lemma We

b e as

the

IU/F

and

~]L+ { ud~f)~

e = c -1

Let

F

by

b e an

in

such that

is a s t r o n g s {

e

from Let

~+

c

We

claim

to

45 .

be an

c(i,eu):u

EHMTI]7.12.

a E A ).

u d -I c ~ c I s ( ~ + , ~ )

in t h e h y p o t h e s e s .

function

:

ud and

Then

induced

ga c u d ( a ) . 6 F.

For

Now

all

for a l l that

i<~

g:ud.

4(i)

: i<~ } C a }

and

E F,

follow

kEpq

we have

c F.

7.6 w e h a v e

conclusions

c F},

suppose

i<~

(ki)j=c(i,qi)j)}

other

U.

: i<@ > e a } E F

: < c ( i , q i ) j : i<~ } C a }

:
[HMTI]7

base

we have

: {jCl

: i<e } ca}

by

let

Let

be as

aCA

: < (kij)

Now

Moreover,

base

ext-base-isomorphism

Let

{jcI

:
with

uEU > .

with

{ ud F

Cs reg

everything

a lemma.

csregNLf

ud~

: iEI } , ~ ) - c h o i c e

uCU.

we prove

: (3kEPq){jEI

sub-base-isomorphism. (
First

ki/F

Hence so

qEga,

{jEI

:

: c(i,qi)/F,

by the

Oi+ c Cs reg. from

and

regularity

as d e s i r e d . Also

~(~U)

=

[HMTI]7.12.

3.5.1.)

continue

the p r o o f

of

3.5(i).

Let

e~

and

~w.

Z

denotes

3.5.2.1.

the

set

U

as

183

of

integers.

follows:

d x = {qe~u d y = {qe~U

: ql

Let

Z•

and

and

zEZ.

w

uEU

U d Z• Then

We u§

define

+

d
: U•

u(1)

> .

= qo +1}'

: q0(O)

Q d yU{qCeU

K d Sb

:

e qo(1)O}

,

qo(O) 2 > max{z 2

: ZCqO(1)O)

and

q0(O)

is o d d } .

~d= ~( ~U){x,y }, ~{ ~<~u){ x,_0~ Clearly,

x,y

and

elements.

Therefore

= base(Q)

= U

no

Gws

sub-base-isomorphic

c.

= b.

in

IUI

3.5.(i), the is

as

in the

and

~+

and

ud E Is(0[,{[+), 6~+

and

b + = ud(b). Then and

Thus

zEZ.

Claim

Then

3.5.2.

To p r o v e Claim

b

this,

~)~ ~

that

every

of

U+

~.

A+

use

h E

~

x , s 2ix

~

: nCw

and

Let

_< d12 A(O[)h(x)

(H,y) )>

such

qEb

shall

the

as

obviously

Let

are

U + d= W U / ~ .

~.

{)[+ = ud*6%

= base(~5 +) the

the

follows:

= U+

universes

notation

B + = S g { x + ,Q + ].

~ U+

is

following on

use

Let Let

gE~U. g~U

> /F.

two

other

Then and

(VqEh(x))q0(1)

A(~)x

= 2. and

Let

to b o t h

= 2.

Let

bEA

qo I ~ qll.

k

~)~ a n d

claims.

:d M =d ql I # q01.

: iE~ ) /F. that

we

there

Since

sub-base-isomorphic

Is(~,~).

Suppose

and

base({]L+) B+

=

3.5.(i).

3.5.1.

and

and

establish

Let

z+ d << z , < H U { i } , y

permutation

and

of L e m m a

: U+•

Gws

first

: 2.

b,s21b ~ di2.

+

is no

_< d12

A(~)b

{)~

ultrafilter

b s AUB

d < g(n)+z

we

have

[ ~

shall

nonprincipal

define

There

h(x) "s~h(x)

such

let

We

we

ud E Is(~,~5+).

g+z

3.5.2.1.

Proof.

3.5.(i)

A + = S g i x + ,y + ~

We

that

Proposition

regular base({)~)

show

to b o t h prove

[HMTI]4.2.

@i+, ~5+ e c s r,e g N~ L f

For

~ d g/~.

of

dimensional

by

shall

formulation

Then

~+.

We

will

a fixed

ud

finite

csregNLf

this

F

= ud*~.

locally

= ~.

proof

Let

of

be

are @l,~e

and

Throughout notations.

Q

For

: U+ >~

= qi(1).

Therefore be

~b

arbitrary

We

show

that

every

zEZ

U+

a

k : {< s ( z , M ) , z + ) ,< z + , s ( z , M )

be >

:

164

3 .5.2.2.

: zEZ} u R]Id, ky+:y+ ,

Now iff

for some

of

ud

and

(and hence

by 3.5.1).

Also

(p(O):z + Therefore

pEb + by

by

pey +

we have

iff

kopEy +

p(0/E(z,M)) since

pEx +

then

Ay+=i

and

y+

from the d e f i n i t i o n

p(O/s(z,M))Cy +

: zeHU{j}}EF,

C y+

iff

{jew

: zEH}eF,

so the equivalence

(p(O)=E(z,M)

iff

follows.

p(1)=s(z+l,M))

p(1):(z+l) +) .

A+I~ c Id

q6b

q11=M.

{jew

since if iff

exists.

To see this equivalence,

iff

kx+=x +

k

pC~U +

it is easy to check that

p(0/z+)Ey +

and

Such a

since for every

p(O, z+)ey +

is regular,

R.

and

by

A + : Sg{x+,y+}.

k(Po) =po:E (qo)

by

Let

qol#M

P d coq.

and

Then

k(Pl) #pl:e (ql)

kopeb +. Thus since @i+ is regular and Ab+=2, i + 2 9 b +.s2b -dl2, and hence ~J~+ b b +'s2bl + { d12. Pkpl

we have Therefore

Also,

~

b

b's~b ~ d12

by

ud E I s ( ~ , ~ ) .

QED (Claim 3.5.2.1.) Claim 3.5.2.2. E i s ( ~ , ~ V ~ ). Proof.

Let

Let

V _c ~U

Then V _c ~U

rl v E Is(~,~).

Let

be a

Z ~(rlvx)=1~ ' Co(rlvx)=c

Then

by

Gws -unit and let

EV3(x~V) E CI

Let

~

~ V ~.

(m,M >EW.

qEV.

~

We have

and this means that by

implies

Assume

d q =

q~Ex

and

u~W

~

c0x=clx=1. : V.

q~ E xnv by

q~E\7.

that

It is enough

{<m-1,M >,<m+l,M > } ~ W. Let Then

rl V E

5 W.

_EV3.[xAV) : C ~V3(xAV) so UO

qCV

u = (m-1,M)

Z•

~W (P) E Subu(V).

p(0/<m,M ) ) (1/< m,M ) ).

E C-[V3(xnV) O

Gws -unit and assume that

(VWESubb(V))(3MEK)

to show that this implies _

be a

Therefore

Then

q E

for some

u.

Similarly,

q E

< m + 1 , M >EW.

QED (Claim 3.5.2.2.) NOW we prove Assume

that there is a

Then there is an that

3.5.2=3.5(d) (c). Gws

~eGws

rl V E Is(~,~)

rl(~V) e Is(~,~*~).

sub-base-isomorphic with unit

and there Let

is

V

to both

and with base

f : y >4 U

h ~ rl(~v)-ioforlv 9

~ y c U

and

such

such that Then

~

h E Is(~,~).

.

165

3.5.2.

By C l a i m that

3.5.2.2

(3 LEK)

_c W.

that

and

3.5.2.1.

Let

c_ Zx{L},

zeZ.

Let

therefore LEK

be

9 VnQ]

N ~ {uEY'

: (3qeV)q~

9 V~Q_}

the

since

definition

regular

and

of

~

Q

n

: w >-

EZ) l{iEw order For

and

even,

= "least

let

wE f*N

Q9

that such

that

Then

Let

y,

d Zx{M}.

Let

by the

and

and

AQ = I.

definition

in t y p e

mappings

t

and

~.

ITI=INI=m

Since

h(Q)

of

h

(VuCf:~N)q~E-h(Q)~.

Such

that

c Z•

Y' : Z•

t i : "least

such

be

show

f(z,M) ( 1 ) : f ( z + l , M ) (1) by Claim

f~(Z•

be two o n e - o n e

f~:N

to s h o w

eW (p) 9 Subu(V)

is r e g u l a r

: t 1.+z:n.}I<~.l

f*T i

f*N

We

and

: i,

(VqE~U)E(VuCf:':T)q~Eh(Q) and

that

and by

A(~)h(Q)

It is e n o u g h

Let

element

: (3qeV)q~

I 9 Subb(V).

q d p(O/< z,M > )(1/< z + l , M ) ) .

such

the g e n e r a t o r

N = y'NT

9

foq 9 h(x). T h e n

T { {ueY'

Then

(3MeK)Z•

= f(z+l,M)(i).

Let

q 9 Vnx

Consider

have

f;~ (Z•

(VzEZ)f(z,M)(1) Zx{M}

we

n

Define

element

ti

of

(~zCZ)[IzI~i

~

t

that

exist.

is

we h a v e

Let

such

E A

: ~ >~

and

f*T

(Vz 6

In fact, ni

f*TN{tj

by

well-

inductively.

: j
t.§

For

ni :

i

odd,

1

interchange Then

the

roles

[,~ E U+:WU/~.

the p r o p e r t i e s

of

q(O/n)~h(Q)+l. T~N

of

= 0

by

T

~+ t

Let

and

e CS[ eg

and

n

~ ~ {E+z

(VZEZ)I{iE@

u

of

(U§ Since

t

we h a v e

= y

such We

show

n

are

3.5.1

and

~ ~ n/F.

therefore

: zEZ}

and

N ~ {~+z Let

d

l

that

and

(~qE~U+)Eq(O/E)Eh(O)

: t.+z=n.}l<~.

one-one

A+I~ and

+

: U + >~ )

by

and

: zCZ}.

d : {<[+z,[+z),< n+z,[+z

that

: (3qe@U)q~Ey}I<~

{ ~ t/F

by L e m m a

Then be a

U+

: zEZ}

u

c Id.

RgtURgn C Z•

(VqE~U+)(VzEZ){q(0/~+z),q(O/~+z)]

I{ u T z • y+

and

U+

Let

we h a v e

1

permutation

N.

for

N y+ = 0

by the d e f i n i t i o n

of

some

L~K

since y.

Therefore

+ .

By the

definition

of

x

we have

( V q e ~ U + ) ( V u , v e u + ) < q.u v01~ ~X

+

iff

3.5.3.

166

v = u + l ).

Let

v=u+1

iff

By

By the d e f i n i t i o n s

d(v) =d(u) +i.

A + = S g { x + , y +}

=h(Q) +.

Now

contradict

Claim

~ x+

t h e n we h a v e

This

of :

T,N X

+

and

and

we have

Therefore

q ( O / n ) ~ h ( Q ) +3

contradiction

~ h(Q) +=

and

establishes

d(t)=n 3.5.2.

3.5.2.) ~

Let

M c Z.

~ ~.

H d {hEZu + : h

is o n e - o n e

and

(VzeZ)h(z+i):h(z)+l}.

Define

GMA =d {hEH

: ( V z E Z ) ( V q E a U + ) E q ( O / h ( z ) ) e v +~

iff

zEM]} .

GMB =d {hEH

: ( V z E Z ) ( V q E ~ U + ) [ q ( O / h ( z ) ) E O +.

iff

zE}i]}.

Claim 3 5 3 1 Proof.

d

.

A + I ~ c Id.

(VqE~U+)[g(O/[)~h(Q) +

3.5.3.

Proof.

Therefore

d h(Q) + : h(Q) +.

QED(Claim

Let

U , v E U +.


For e v e r y

nCw

define

L

~ IzEZ

: z2~n2}.

Then

LeWSb

n

is s u ch that

URgL:Z.

arbitrary. Let

Let

McZ

t ~ pjoOk.

be fixed.

Let

Z w

Let

T ~ < {t(n)+z

kC~(wx~)

: zCMnL

be

}U{ (t(n)

+

n

+n) 2~

:

new > .

Then

T e wSb Z.

Let

h(k,M)

d < { (t(n)+z,

w

,( T n , k ( n ) i > ) : new > /F EZu +

is a o n e - o n e

:

mapping

zCZ > .

Then

such t h a t

h(k,M)EH

(VzEZ)

We s h o w t h a t

A B h ( k , M ) E G M n G M.

Recall

elements

and

Ay + = AQ + : i.

Let

~ c ~ U +.

implies

Z~Ln,

Now

l{n~

i.e.

q ( O / h ( k , M ) z)Cy + I{nEw

e n o u g h to s h o w l{nE~

We

=

h(k,M)eG

z~M.

l{nE~ .

Then

y

h(k,M)(z)+l +

and

Let

Q

+

zEM.

h(k,M)C

= h(k,M)(z+l). are r e g u l a r

Then

: n2
t(n)+z{T n

shows

q ( O / h ( k , M ) z)~y +

since

: t(n)+z:(t(n)+n)2}ISl.

N e x t we show

h(k,M)eG

( ~ z E Z ) q ( O / h ( k , M ) z ) { Q + ~ y +.

: (t(n)+z)2

Therefore Let

Suppose

: t(n)+Z~Tn}I

Therefore

z2>n 2.

that

since

> (t(n)+n)2]l<~

shows

Let

.

By

y ~Q

it is

zeZ.

Then + + q(O/h(k,M)z)~Q ~y .

h(k,M)EG~.

k,dEW(~x~)

show t h a t

be such that

h(k,M)#h(d,M).

Let

{iE~

: ki#di}EF ,

D ~ {i~m

i.e.

: k.#d.}. 1

l

Then

k/F#d/F.

167

3.5.3.2.

(VnED)(k(n)O,(Tk,k(n)l) # h(d,M)O.

By P r o p . 4 . 3 . 7

of

ECK]

ultrafilter

on

~

nonprincipal =

IGB]

we d e f i n e

3.5.3.2.

Proof.

an e q u i v a l e n c e

L -: M

such

that

h(k,M) O #

w

since

every

Therefore

M = {r+z

IG AI

entirely

such

that

Then

L ~ M.

Then

G B"

for

Rgh = Rgk

we o b t a i n

shewing

Z.

by

h,kEH.

Let

this

for e v e r y

Then

zeZ

feZ

and

q(O/k(r+z))~y ~ ~

M - L.

The

prOof

for

be fixed. qeaU +

r+zEM.

GB

is

analogous.

QED (Claim Let

be

Then

A GL

L,MCSb

: raM}.

q(O/k(r)+z)ey + ~

: rEL}

Let

h(O)=k(z)=k(O)+z.

of

rEL ~=~ q ( O / h ( r ) ) E y + ~

Sb Z.

Similarly

Rghr]Rgk # O.

zEZ

-= on

L , M 6 Sb Z

by the d e f i n i t i o n

Then

I~/Fi=x

~ -regular.

(3zEZ)L={r+z

Rgh(]Rgk = O.

Assume is

iff Let

(Vh6S A) (VkEG A)

Then

is

have

shows

3.5.3.1.)

N o w we d e f i n e

there

we

}

= x ~.

QED(Claim

Claim

} # (d(n)O,(Tdn,d(n)l)

3.5.3.2.)

W C Sb Z

Sb Z / -:.

For

be a set of

every

LEW

(~{hEG A) (3k@GA+)Rgh=Rgk G LB+ _c GLB

similarly.

representatives

let

and Then

A+ C G AL GL

for

the p a r t i t i o n

be s u c h

that

(kfh,kEG A+) [Rgh=Rgk ~ h=k3. B IGL+I

still

A IGL+I

=

= xw

We d e f i n e for a

(since

I

given see

hCG L

there

the p r o o f

of

Claim

3.5.3.3.

Proof.

Let

Then

LcZ

rEZ.

Let

A+ c GM follows

entirely

(~ueU+)(~!hCU{GL Let

hence

k d (u+(z-r)

such from

QED(3.5.3.3.)

many

keG A

with

of

define Then

: zCZ ) . This

: LeW})UeRgh.

and

(3!MEW)M-L.

definitions

analogous.

A+

qE~U +

Rgh=Rgk.

that the

countably

Rgh=Rgk

3.5.3.2).

ueU +. and

are o n l y

Then proves W

and

The

L d {zeZ M = {z+r

k c GMA .

A+ GL .

The

G B"

: q(O/u+z)~y+}.

: zCL}

Then

existence.

for

same

for

there

some

is

h E

Uniqueness proof

for

GB

is

168

3.5.3.4.

For

every

function. >~

let

: LeW}

A+ : GL

gL

P d U{g L

>

:

3.5.3.4.

R

:

B+ GL

>~

: LeW].

is a o n e - o n e

R d {< h ( z ) , p ( h ) z

Claim e

Let

U { G B+

Let

L~W

Then

and

: U + >-~

U+

: U { G AL +

p

onto

h~U{G A+

be a one-one

funtion

: LEW}

and

: LeW}

>~

by Claim

and

onto

3.5.3.2.

z~Z}.

is a p e r m u t a t i o n

of

U+

and

~ c

IS (~t+ ,~+) .

Proof.

Let

hCU{G A+

: LeW}

< u,v )CR U +.

uEU +

iff

U+ show

arbitrary.

argument, with

domain of

that

and

U +.

Ru:Rv+l

definition

Let and ~

by

q E ~ U +.

Let

R u : k ( z ) ]. q(O/k(z))EQ

regular

R c

QED (Claim

:~ and

ud*~=~

Lemma [l~I=y

Thus

by

Then

with

domain

shows

prove

~ E

that

is(~eU

By Lemma

that

: U + >~

+, @~eU+).

3.5.1

since

R

R -I

we

have

u=v+l

iff

R.

(3L <_ Z) ( ~ h e G A) ( 3 k e G B) (3zCZ) [ u : h ( z ) A GL

of

q(O/Ru)~Q + ~ Therefore

v E U +.

we

RoqCQ +

R y+

have

q~y+

since

y+

*=~ z e L ~

and

Q+

are

= Q+.

i + : Sg{x+,y+},

B + : Sg{x

+

+ ,Q }.

~ E

i s ( ~ + , ~ +)

+.

Then

~ .

is a b a s e - i s o m o r p h i s m

~+

between

is e x t - b a s e - i s o m o r p h i c

Actually,

udB-lo~oud A E

ud~'~:

to b o t h

#~

Is({~,~).

3.5.3.)

these

Proof

Then

G A,

is a u n i q u e

3.5.3.4.)

and

QED (Claim

of

definition

Is({~+,~ +)

that

.

U=qO. the

+ ~

have

~

By

By

elements.

Thus

We

the

of

R x+ = x+

Therefore

Let

statements

: 0_+ .

there

a function

Therefore

R y+

.

is

instead

These

x + = { q ~ C~U+ : q 1 = q o + l }

3.5.3.3

u:h(z). R

GB

U +.

is a p e r m u t a t i o n

holds

Let

Therefore using

R x+ = x+

By C l a i m

UCRgh.

that

v = p(h)z.

is a f u n c t i o n

We

such

A similar

>-~

be

of

3.5.4. and

claims,

Proposition

3.5(ii):

We

shall

Let

~=[ylk~.

(VVC_I~]EIVI
3.5.(i) need

There

is p r o v e d .

lemmas are

rlv~lS~].

3.5.4,

~]L,~EBA That

is,

3.5.5 with ~(~ is

below.

~ < not

such ext-

that

3 .5.5.

~@~

-isomorphic Proof.

to any

Notation:

Boolean

~:I~I

Let

Thw

~(T+)

{y~

and

Let

~ ~

~B

BA ~ ( ~ )

be BA-freely

V~T +

generated

For each

generates

with

IVl!y.

vny =O.

completes

Let n.

~(~)G.

new

Let

for some

Thus

~ ~ ~(y) by

{x

~Cy +

some

with

: ~Ey +}

let

v ~ x ~.

o~ ~ ~(y+),

since

Then there is an rlvqIS~.

~ey+

Observing

the proof of 3.5.4.

and

~, ~

~ c ~J~

Then

with

IL I<w

Then : aEL >

:

Mne

there exists

ha

(( ha(Pa ) : aEL > F d < Eiha(X,a) =ha(a):f(a)

and

Gco.

be two

and f C_ F. Let

Hence G --w c B

LCZd~)~.

Since

:

~L=I ~

Let

: aeL}

proves

[~(~)Rgf. Let

~ d [~(~)~)L and

d L : zd

we have

and Let

aeL.

Then

n, hence by EHMT]2.5.25

e iS(Pa~L ~ a ~ , P a E L ~ f a ~ )

and hence

(~':) below.

(:.~) (VGC_ B) (3F e Ism( ~ 0 1 ) G , s

g~(0~)Dof ~

By EHMT~O.3.6(iii),

: xcQ > c Is(~,Z). L1f _c F

Then there

by EHMT]2.4.7

are both of characteristic

: p~paCL(RIa~)>

both of cha-

be arbitrary.

xER > C I s ( ~ , P a E L ~ f ( a ) ~ ) .

e is
CA -s c/

f c Ism(~5,~).

xCQ > c Is(~,pac L ~{~a~)

: aeL >

~fa~CE

statement

to be the unique

We base the proof on the

(:'{) there exists an

Assume the hypotheses.

< <x,f(a) ~a~'

By

and hence

~(s

< < x,a

~(H)

see HausdorffEH].

BA-freely

FEIsm(~(~A)B,/[)

Proof.

Sb H.

(VzEC~{O}) IzI=y +.

: ~ey+}

3.5.5.

exists

we define

3.5.4.)

racteristic

d

H

Then

_

V~

Z ~(y+)BA QEm(Lemma

B=2 x .

~ C ~(y+)

IzI>y.

such that

below,

be a cardinal.

ICl=y +

(Vzs

d At

~ li~l 9

(VxCA~{O})IxI=~.

BA. Let

Clearly

(~) and

such that

Lemma

I~[

set algebra with universe

Let

with

with

For any set

well known result (e)

~

c_ F.

Let

aEL.

G1f _c F.

Then

and hence F(a) =

We have proved

170

3.5.

By
statement

E ism(~(~)B,s

with

(*) implies the existence of

f c F.

3.5.5.)

QED(Lemma

Now we turn to the proof of 3.5(ii). T=IyI~+w. ll~l=y

By Lemma

(VU!1~)[iUi
Define

Let

that both

Assume ~

: yExJ ~@ f(l ~)

~

and

~

9(~)k*B

~

that some

. Then

there is a e Is~ .

derived

~ ~ ,

VSI ~

Let

Crs a

with

and

k { < U{a({y}•

and

~ { ~[ k(i~).

and

~,~

Then

E ~Gsa"

with

since

: yEx}

:

f E

There

n. Hence

by Lemma 3.5.5,

~

is

n

such

~ ~Z ~ f*~

~

-isomorphic

to both

~

C y

to both

and and

f

(c~,)

and

A•

~

By

that

Hence no

xAU#O

since

~

above.

We have

is sub-base-

Crs

is sub-base-

.

The b a s e - r e l a t i o n

~,~ECrs

is defined

to be

fAB ~ {< x,y >CA•

:

Now:

with

Let

rlvok E

IUI~]base(V) I~y,

formulated

are lower base-isomorphic f

then

rluEIS~.

function and

By (**:=) and by the above, Proof of 3.5(iii) :

rlvk(X)#O

Then

Hence

~.

(f-~)y c x}.

-one function

of

~ .

and

on

~

If

rlvEIS~.

from the assumption

be a one-one

induced by

and

Therefore

property

a contradiction

f

is sub-base-isomorphic

U { (pjo)*base(V).

to both

~(x)

: xEi >

Ibase(V) l~y

-isomorphic

:

Let

~

Ibase(~) 1~]base(~) 1:Ibase(fl~) L=11~I ,~=y,• Y.

this contradicts

fAB

set algebras

are of c h a r a c t e r i s t i c

(3yEx)Vn~({y}x~)#O.

Let

~>0.

~.

k~i

then

and

below holds.

k E Ism(~,~)

~(~)feA

a>l

~ rlu~IS~].

~ {

E Ism(~,~),

and

(**)

f ~ (U{a({y}•

xEB ) .

Let

3.5.4 there are Boolean

and such that

(**)

:

F E

~

iff

there is a one-

fAB e Is(~,~). and

5~<w.

~

are not lower base-isomorphic.

Define

V ~ ~2Ua{2,3}u~{4}U~(~-~5)

171

3.5.

and

W ~ ~2u~{2}u~{3}u~{4}u~(~N5).

~/~(~[W).

Then

~,~E

and it is obvious Let

~<5~

Gs~ eg

that

and

~

~

~

by Prop.4.2,

~

and

Let

~

(~V)

are not base-isomorphic.

/3[,~E Gs~egmLf

,

~t~.

Suppose

V~E{{4},{3,1},{2},{2,1},{I}}.

V~=V~.

If

V~={4}

then

~,~ECs

V~:V~:{3,1}

then

6~,~

are base-minimal

Let

If

IAt Zd AJ=2

and we are done. E Is~

Suppose

and similarly

E Subb(~))~(Gy)~ similarly

for

~(GW)~.

V~E{{2,1J,{I}}

~

IAt Zd AI:I.

~(~W)~.

By

~=4. ~

and

~

Then

(VYESubb(~))rl(~Y)

~(~y)~E

Cs reg

by

~c

Gsreg~Lf

Hence we are done by EHMTI]3.6.

Let

~w.

Y ~ {fEe3 ~,~63Cs

: HIfEH2}. nDc a

by

I@~HI2~

1.3.

defined

so that

KCAx

x

Then

seen that

x

~{

X { {f6~3

~(Z){X}

~

and

are regular.

~ 6Gws n~

is regular. and let

be such that

IHI~b~HIk~.

: HIfEH2 (p)}

and

~ { ~(~){Y}.

Then

and by [HMT32.1.7.

and

Clearly,

be infinite

Ax:H.

~

HC~

Define

We let

Now we show that Theorem

Let

and

The cases

Proof of 3.5(iv) :

~ { ~3.

E

(VYCSubb(~)) (VW E

Hence

are similar to the above ones.

and

If

are base-minimal,

~<4

p ~ H•

we have

and we are done by 3.4(2).

and

Let

Then

and we are done by [HMTI]3.6.

then

for ~ .

~

by [HMT32.5.25

there are 5 cases:

V~={2].

and

Let

To this end, we use

x6{X,Y}.

Now we check that

FC ~.

Let

i g2{c(F )x

for any

is small.

Then T h e o r e m

ieK~F.

g6~3. 1.3

x

Then

Hence

Now

x

is small. iEH~F

that

~

Let

since

clc(F)X=O.

implies

was

We have and

are regular. Next we show

~ ~.

Let

xE{X,Y},

q ~ ~3 (q)

to show

is

Q-wsmall

by Remark 4.10 since we have seen that

Let

F--~ c ~,

q[F/s~ex.

and

SE~3.

Thus condition

Then

"

and

use Prop.4.7

fEx

rlo e I s ( 9 ( s

q6x

f[F/s3Ex

Clearly

iff

A(f)Q:O

since

l{x}l=l.

~W(~)

EHMTI35.3)

and

IAxI=IHI~.

Then

is simple by

x

rlQ~IS(~,(i){x}),__s

x

iff

Cond.(i)

~eCs

and

is small.

s*(H~F)C~_

(ii) of 4.7 is satisfied.

satisfied

We shall

is

(see by Prop.4.7(II).

172

3.6.

Let

q ~ ~•

Then

rlQ*~=rlQ:~

by

It remains

QNX=QnY.

to show

f : 3 >~

3

clearly,

Z#Y.

we have

For

EIs~

(by

and

and

rlQ ~ I s ~

Therefore

that

@i

is such that

seen that

W(f2)Nz=O

rlo E I s ~

and

~ ~5

~

rl(Wn)

n<3

let

~ Is~5

W2AX=O).

Then

W(fo)AZ=W(fO)Ay.

by the above.

.

are not b a s e - i s o m o r p h i c .

~ ~ Is(~,~).

every

,

Let

Z ~ ~X.

Then

Wn { ~3 (< n : i<~ > )

for

nC2.

fOE2

and

Hence

A contradiction,

and,

since

Thus

since

ZEB Above,

f2=2

f~2c2.

Suppose

rl(WfO)e

Z#Y.

Q ED P r o p o s i t i o n ( 3 . 5 . )

The algebras uncountably

Problem

~, ~

many

3.6.

in the proof

~aw.

Do there

isomorphic

~,~EGsregNLf

they are not

lower b a s e - i s o m o r p h i c ?

implies

that

weaker With follows noted

generated

~ <~

two c o u n t a b l y

all

subbases

3.7,

~u=~

by

generated

finite

3.8 below.

cannot

3.7.

]aI~x

for n o n - d i s c r e t e

(in [HMTI33.19)

is an

were

such that

Prop.3.7

be r e p l a c e d

with

below any

in EHMTI]3.18.

IAI~

be such that

there

to P r o p o s i t i o n s

the c o n d i t i o n

from

exist

both with

r e s p e c t to the c o n d i t i o n

Proposition 1

refers

condition

3.5(ii)

elements.

Let

EHMTI]3.19

of

~.

to be n e c e s s a r y

Let

U

in 3.7,

and

~EuCs~egnDc ~

J~1~<

The c o n d i t i o n

it JAI~

Let the c a r d i n a l s

U

such that

that

was

in [HMTI~3.18.

be a cardinal.

w~l~1~I +

note

Assume

~
(i) - (ii) b e l o w

x

~#x.

and Then

hold.

+ (i)

JAe~x

(ii)

For

any

and WcU

and h e n c e

l=u{ IAxl if ~

: xEA}.

IW]=~

then

rl(~W)

is not e x t - i s o m o r p h i c

~ Hom(~,~) to any

for any Cs

.

CrS

3.7.

173

Proof.

Let

i~l=v
such

:iv-~Hl=v. K

by

1

v#~.

~
n

be as

exists

: HU'

because

: qO=u}

:

uEm}.

cardinal

~

such

qO : n(H1q)

and

y~ d= [q~ ~U d y = {yu

IL l=u.

be

and

such

such

that

and

there

O~H

IHI =

Such

a

U' ~ U~{k}.

Ine(HK) I>•

and

is

and

IKI=~. let

that

= ~v>~

Then

that

be f i x e d be

IHKI

T { {{qC~U

v~L CH

KcU

~ U'

:

every

be such

k e U~K

d x = {qE~U

For

Hey

Let

Let n

in the h y p o t h e s e s .

Let

~ ~ ~U.

K c U'

function

and

that

Let

exists

Then

U,~

IU'I>~.

Such

a

We d e f i n e

HIqcHu'}.

that

v<~<~

let

L <~

be

such

that

Let

: ( V i ~ L )uq i = k } .

: v<~
and

~=Iul}.

G ~ {x}UTUY. Let

6~ ~ ~ ( s

Clearly

~Dc

since

(Vz~G)v~Az~v~(HOl)

and

iv~(HOl) I ~ . First every

we

show

element

x 6 Sm 6%,

Thus

of

too.

q E e(F)x .

G

v
~

Then

we h a v e

[Ay

O{IAal +

: a~A}~l.

Then

a E Sg G O

for

t h e r e is a c a r d i n a l

(For

the n o t a t i o n

E Su~, iAI_<~.

i.e.

Next

G o C G.

~
such

Dm

since

show

some

see

IAa~l<~.

]GI=~

we

=

iKi

~

We

i ~

(HU1)~F.

IAe!~.

For

Y~G

and by

finite

(Vg6Go)Ag~, Then

every

Let

gYm

-< i~I

cardinal

IAxI=~

we

Let

aEA.

subset i.e.

aEDm

It r e m a i n s

and

that

: f*(HUl)CU.-{k}} _ .

every

0.)

Clearly,

show

(~aEA) IAal
IAaI~u
1.3.

~L is regular.

by

that

that

section

Then ITi

and

For

Thm

and

that

Therefore

use

Toy ~ Sm ~. FC e

have

: a~A},

i=i~ul= u.

shall

x _C { f E ~ U

1.3 we

1 = o{ iAai +

we p r o v e

Let

since

by T h m

We

Clearly,

Ax = HU1.

i )x qk~c(F

Then

have

is regular.

is r e g u l a r .

Clearly

x E Sm ~.

Next

G

that

GO

G o ~ D m ~.

by

Dm

to s h o w _< ~

of

b~7

E that i

_<

+ <

lal

Therefore

It r e m a i n s

IAI <

to s h o w

Hom($][, ~ ( e W ) ~ ) .

Let

that WcU

~uwul~l

= ~,

for a n y be such

Wcu, that

by

[HMT]O.I.~9,

if

IWI=~

IWi = ~.

then Let

0.i.20, rlA(~w)(~ V deW.

174

3.8.

Let

~{

~ V ~.

Suppose and

u ~ K~W.

CoZ=~.

Let

However,

rlv(coZ). that

First we show that if

Thus

HK ~ {H1q

eV~CoX

such that

c~V](rlvx ) showing

{~e

K~_W.

Clearly

n(H1q)~,

by

by

(~f~W~x)H1f~.

3.8 below

Assume

KC~q.

CoX= { q ~ U

KCw~U'._

by

uEK,

EV] c O r• v z #

and hence

We show

: HI qeHu'}.

Thus

there is

q6

In~(HK) I > ~ I W [ . Therefore ~ . . Therefore q •

shows that the hypothesis

from CHMTI]3.18 isomorphism

Proposition

if

z~TC_A

q EV] 0 (rlvx)

3.7..)

Proposition

ordinary

u~W

Then

rlv~HOm(~,~).

rlv~HOm(~,~).

QED(Proposition

be omitted

by

: q~eWnCoX}

since

that

rlv(Z)=O

# rlv(coX).

then

: qO=u}.

rlv~HOm(~,~ )

c~V](rlv(X))

Therefore

z ~ {q~U

K ~ W

3.8.

even if we replace

~u=~

cannot

ext-isomoprhism

with

or homomorphism.

For each

csregADc

~
of power

~w

and cardinal such that

Is[

~>~

there

(~EGs)

is an

[Hom(~,~)~O

~

(VWEsubb(~))IwI>~].

Proof.

Let

HC@NI

saw

be such that ~ d= ~ U

Let

and let

since

U

be a cardinal and

IHI=~

Then

implies

< -

that every

6 GS.

Let

d (~) and -

X ~ {qE~U

IAI=[~L

Let

: qO~q*H~. ~j~ECs nDc

and

subbase

of

Y d= h(X).

h ~ Hom(~,~)

~Gs ~

Then

the hypothesis has power >e. (~iEHNI)y

Suppose

there

Hom(~,~)#O

Suppose

-< -d(~)Oi

h : {~-

since

is a subbase

W

(~ieH~l)X~ of

~5

O i

such that

IWI~.

(Then

Then there exists NOW,

U>e.

AX=IUH.

Next we show that for every

~

Let

I~HI~.

~ d= [~ (s

and

such that

q~c0Y

Therefore

a

since (~)

co

.

~

y~l.

qe~W~

W~O

since no subbase such that

(~w6W)[(3iEH~1)qi=w But we have

(~

Co)X=l~

is empty

qe(H~1)=W, and thus (by

if

since

~#O.) ~eGs

.

qwO~vc ~ - - d(~)] Oi "

IUI>~),

contradicting

3.9.

175

h~Hom(~,~). every

We have

subbase

It remains in section

of

seen

~

that

Hom(~,~)#0

and

~EGs

imply

that

is of p o w e r >~.

to show that

Reducts,

~

is regular.

in Prop.8.24,

6 ~ E C s reg

will be p r o v e d

because

it uses m e t h o d s

remains

true

of that

section. QED(Proposition

3.8.)

We c o n j e c t u r e "IAI~" by

is r e p l a c e d

~

elements

Conjecture = Sg(~)G Assume

Let

K E {csreg,ws

~ = ~
~u

to some

that

condition

(VxEA)~IAxI".

lwOGUSl

< • _ < _

K

}.

Let

Ibase(~) I.

the above

with

" 0g can be g e n e r a t e d

In p a r t i c u l a r :

~EK,

Let

Then we c o n j e c t u r e ~E

if the c o n d i t i o n

that

SCbase(~)

A =

i d = U{ iAxl +

: xEA}

6~

is strongly

ext-

SCbase(~)

conjecture

is i n t e r e s t i n g

only

in the case

~
Proposition

3.10(i)

is an a l g e b r a i c

to the e f f e c t elementary

in 3.10(i)

statements

version

extensions. model

Note

that

theoretic

3.11,

3.10.

Assume there

is a ~

equivalent

model

result

since

Proposition

theoretic

structures

(i) of P r o p . 3 . 1 0

not only

~,~ECrs

below

~,~E

[HMTI]7.30(g).

have

below

theorems isomorphic

is stronger

the e l e m e n t a r y

isomorphic).

In this

extensions connection

3.13.

Let

(i)-(vi)

in

of the various

are identical E n d

see also P r o b l e m s

Proposition

b e l o w was q u o t e d

that e l e m e n t a r i l y

than the q u o t e d

(i)

by the w e a k e r

and

We note

3.10

[HMTI]3.18

and

3.9.

isomorphic

when

that

CS Cs

be such that

~ .

Then

hold. and

base(~)nbase(~)=O.

e__xt-isomorphic

to both

Let

~w.

{7[ and

Then ~

.

176

3. i0.1

(ii)

Assume

~,~csregnLf

and

~.

is e x t - b a s e - i s o m o r p h i c to both (iii)

~ECrs

strongl~ e x t - i s o m o r p h i c

KE{Gs,

G w s , Gws nOrm~ , Gws~ d, Crs reg}~ .

Assume

~EGws~

Omp

to both (v)

(vi)

~

and

~EGws~ ~

and

Proof.

~

Ws

is e x t - b a s e - i s o m o r p h i c

Let

hEIs(~,~)

Lemma 3. 10.1.

There are isomorphic

and let

If

~,~EWs

to both ~,~

~

~CK

Let

. Let then

s

base(~)=base(~).

~

nLf

ext-isomorphic

such that no

and

~.

be as in (i). Then

kEms(~,~),

tEIs(~,~)

First we prove a technical

and let

qOCXo ,. .. ,qnEXn .

fEIs(~],~) .

Let

h=kot -1 and ~ E C s

.

lemma.

Let

Xo,...,XnEA

~_>w. and

R d O{qi~.:Axi : i
Then there is a one-one function (VpE1~4) (~i_
~

t : R >- base(Q)

such that

p~f(x i) 3.

A s s u m e the hypotheses.

We shall prove the lenuna by induction

n.

1. Assume 9H{-d. 13

n=O.

Let

: q(i)#q(j)

CA -s,

d(~)EB.

since

Let

d {(q(i),k(i) > function.

Let

AxltoqCk. f(x)EB

d d [1{dij : q(i)=q(j)

and i,jCAx}.

Now

by

~eLf

qEd(~)Ax.

Thus

kCd(~)Af(x) : ieAx}. pE1 ~

Thus

qexEA,

IAxI<w

Clearly,

fCIs(~,~).

and

f]t and

(Existence of partial b a s e - i s o m o r p h i s m s )

@5,~csregALf

Proof.

to both

.

~hw.

(ii) :

Then there is some

with the same base,

Let

Proof of

e#{.

~ Ecs[egALf

~ .

have d i s j o i n t units and

for some e x t - i s o m o r p h i s m s

of

and

base(~)Abase~):O

Then there is

on

~

Assume

(iv)

Let

Then some

Then

and therefore

d(~)eA

f(d(0t)Nx) = d(~)Af(x)

t : Rg(Ax]q) Ax]pCtoq.

by [HMTI31.13

is regular by

is a term in the language

be arbitrary.

be such that

pEf(x)

d

and i,jEAx}.

~ C C s reg.

since

Define

# O

by

t d

>- base(~5) Then

and

is a one-one

Ax]pC_k,

Ax:Af(x),

by

kCf(x),

3. I0.1 .

177

2. Let

new

and assume

that

qOeXo,...,qn+leXn+l

.

permutation

such that

A

and

of

AXn+ 1

and

~w

and

~ ; see

have

~ are

Let

3.10.1

that

iff

O.

Let

be a finite

Such a

T

operation

By [ H M T ~ I . I I . I O , By

9

exists

and by

Recall

By

both

since

~,~EDc in

#]LECs NDc

EHMT]1.11.12(X)

i~n.

Xo,...,Xn+lEA,

D ~ T*AXn+ 1.

is a d e r i v e d

T

Let

we

we have

the n o t a t i o n

f[H/k]

from

We d e f i n e

d Yi = x i ' S T X n + l

d

h i = q i E D / q n + l O T 1].

and

Then

hiEY i

Then,

by our i n d u c t i o n

since

: R 1 >~ base(~) pEf(yi)].

~

with

Let

>- base(Q) iNn+l

Let

poT--IESTXn+I 9

A ( S T X n + 1 ) ~ T * A X n + 1 = D. section

s

~9

n.

Let

T*(AXn+l)nA=O.

by

[HMT31.11.9.

PEXn+ l

for

A ~ U { A x i : inn}.

finite

it follows

holds

is regular. hypothesis,

R I ~ O{Rg(AYil hi)

we have

the p r o p e r t y

that

a one-one

pEI ~

function

be such

function

and let

such that

that

: i~n}. tl :

(VpEl~)(ViSn)[AYilp~tlohi

R = U { R g ( A x i l q i ) : iSn+l}

be any o n e - o n e

and let

Let

t : R >~

(RnR1)Iti~t.

Axilp~toqi.

~

Let

We show that

p 9

9 f(xi). Suppose

first

Ayi~D~tlohi)

i~n.

by

pel ~

and

Thus

p'Ef(xi)

show

Axilp~p'.

Now,

H1h i--qi c

Let

R g ( A Y i l h i ) ~ D o t 1.

Rgtl~base(~ ) .

H1tlohiCtoh i

since Let by by

p, d= p l a y i/tlohi].

Then

Suppose

AXn+llp~toqn+l Now

H { A Y i n A x i.

H A D C--A x i A D = O

i:n+l. and by

Let

p' 9 ~

since

by

: HIp

Now

,

feIs(~,~).

to show

We

H1p~p'. Therefore

Then

by the above

Hip' and by

Axilp~toqi. D1(tOqn+lO~

D = ~*AXn+ I.

H ~ AYoAD.

~eCs

h i : q i [ D / q n + l ~ -i].

(as desired)

We have

f(Xn+ i)

since

A Y i l P ' ~ t l o h i. by

It is e n o u g h

and

Dop'=~

Rg(Hlhi)~Rg(Axilqi)NRg(AYilhi)~RnRl.

p ' ~ f ( y o )= f ( X o ) ' S

D I p ' ~ p o ~ -I

p ' e f ( y i)

p' = p [ A Y i / t l o h i 3 ,

next

Therefore

f(yi ) = f ( x i ) - s T f ( X n + l )

= H 1 t l o h i = H 1 t o h i = H1toqi H : AYinAxi,

Now

Let

-i

) ~ pot

-1

by

p' { ( p o T - l ) [ A Y o / t l o h o ] .

just as in the case H 1 h o ~ q n + l O ~ -I

iSn.

We show

and t h e r e f o r e

:

178

3 . I0.

HIp'

= H 1 t i o h O = H 1 t o h 0 = H1toqn+iOT--i = HIpoT--1

~ R n R I. and

Then ~eCs

QED(Lemma

poT-les

reg.~

EcsregALf

I.

Let

§) = ud AF e Is({J%,~)~

U,W

.

Let

U + -- IU/F.

Let

Then

since

Gi ~ELf

Define

3.5.1.

~ "

since

By L e m m a

Let

such t h a t

(Vis

b

and

qex.

Then

it is e n o u g h definition

ud B

and

{ieI

Then c < bq(J)i

the d e f i n i t i o n

~

in

Lemma

3.5.1.

of

qjER i : j<~ > t..

Then

is

Then

1

: R. >~ U 1

Let

To p r o v e

beW(Iu)

Z d m(<x,Axlq

: weW > . Let

x~B

boqeud(fx)

) ).

Then

Let

iez

bq(j) i -- ti(q(j) ).

by

be

by the

(~ieZ)q ~':(Ax)CRi.

boqEud(fx)

Let

: k~G.} m "

b d
and t h e r e f o r e

: i~Z}

ieI.

t.

: j<~ > e f x } e F Let

_< IIl

for all

(VxEB) ~ ( x ) C u d ( f x ) .

Thus

and h e n c e

by L e m m a

IIL-regular.

pef(x) 3. Let

Then

(VieI) ({ZEE

: iem(k)}

~0qep(boq).

: (bq(J)i

( ~ i e Z ) ( x , A x l q > E G i.

jegx.

tio(Axlq)

b o q E ~ ( U +)

to p r o v e

of

and

N e x t we s h o w that

e d= ~ - 1

and

is

ultra-

ud =

IK] <- IBOWU~I

there

= ti(w).

: W - U +.

and

Ri d= U { R g ( k l )

3.10.i

(k/perU) (V( x,q > E G i) E t i o q C p

Then

F

Let

be a r e g u l a r

#J~+eCsreg

Then

~J[,~E

feIs(~,~).

Let

and

G.! d {keK

is finite.

F

IEl = I I l

such that

and

f(Xn+l)EB

Let

e : U >-- U +, ~

: qexeB}.

exists

Let

Let

as in L e m m a

such that

E

be o n e - o n e .

i~I.

1--W

be

S u c h an

: K >~ E

R.C W

ECF

~_>m.

respectively.

K d {< x , A x l q >

is finite) .

ZeECF

Let

fit >- IBuwu~I.

be d e f i n e d

Let

~ELf

Fix

p'es

EHMT]I.11.10.

is a s t r o n g e x t - b a s e - i s o m o r p h i s m

3.5.1. by

by

to the p r o o f of 3 . 1 0 ( i i ) .

be any set s u c h t h a t

ud-lc_e

A(sTf(Xn+I))~D,

pef(Xn+l)

be of b a s e s

f i l t e r on

m

Then

by

R g ( H 1 h 0)

3.10.1.)

N o w we t u r n

I

f(Xn+i)

by


Thus

: j<e ) e f x

by

ZeF. We have p r o v e d

1

Statement (*)

Since

(~'~):

(VxeB) ~ ( x ) C u d

udof

the f o l l o w i n g

f(x).

is a h o m o m o r p h i s m , computation

shows.

(~) i m p l i e s

that

Let

be d i f f e r e n t .

v,wEW

b

is o n e - o n e Then

as

3. I 0 . 2 .

179

there is then

pCaW

Let

we have

V { ~(i ~)

By (*) we have V~b(x)

pl=w. that

xeB

pe-doi.

b(v)#b(w)

= -ud f(x).

bx c Vnud(f(x)).

b(-x)=V--b(x)

This proves

we have

BI~ =

rlO + = ~ o ( u d o f ) - l e I s ( ~ , ~ * ~ ) .

~ = rlvo(B1b)-leIs(~,~)

(:~)

Then by

Then

By

f(x).

By

is a base-isomorphism.

be fixed.

~x = V~ud

Thus

Thus

b ~Is(~,b:~)

Let

f(-x)

Thus

= rlvoudofeIs(~,~'~). Then

and

= [(~W).

~(-x)~ud

and showing

b*~eCs

~ -ud f(x).

b -I.

Po=V

~(~+) f(-dol ) = --01

bopeud

[HMTI~3.1

such that

Let

d {

is a strong ext-base-iso-

morphism. We have proved morphic below,

to both

is a

Cs reg

~+~Cs reg

~.

Let

~+

bases

If

then

~=0 Let

and

U

~

Then

with

Suppose U=Y).

unit

rlNEIs(~,~)

Y

Let

by

VuW

and with

E Gws c ~

and

~

Crs

fEIs(~,~)

then

and

uny=o.

and

and

W

VNW=O.

Vt~4. Let

VnW=0.

Let

X ~ {xUf(x) : ~

be the

X.

UnY=O)

or

Therefore we h a v e

is e x t - i s o m o r p h i e

U=Y

and

universe

and

V

~ ~

with unit

feHom(~,~)

(~2

by

eCIs(~+,~)

and we are done.

be the full

XESu ~

since

I.e.,

Then there

have units

Assume

~=~

A(~)V = A(~)W = O.

f(x)=O.

fEIs(~,~).

~,~ECrs

respectively.

implies

that either

Then

and

f = (h+le) o~ -I

(iv):

~

3.10.2

of 3.10.(vi).

~,~EcsregnLf

and

feIs(~,/W). Let

xEA}.

ext-base-iso-

We have also proved Statement

such that

(iii)

and have

is strongly

and two strong e x t - b a s e - i s o m o r p h i s m s

~EIs(~,~)

Crs

and

3.10.2.

Proof of

:

~

since

which shows the "(ii)-part"

Statement

and

(ii)

~

rlveIS(~,~) x=O

to both

~ EGws c ~

(~eGws7

with

iff

and

xUf(x)=O

4)[ and base

and

U.

~ .

iff

If

So f a r ,

~,~e (iv)

has been proved. Suppose

= rl~(~Y) and

,~

a~2

showing that .

Let

KE{Gs

r

Then

~ Gws a

rl~

is s t r o n ~ t

Gws ~n ~

r

= rlg(~U)

and

ext-isomorphic GwsWd}. ~

If

~,~EK

I ri W =

to both then

6%

~80

3. I0.

clearly

~EK.

yeC

pE,y, qEVU}~

and

that

Let

y=xUf(x).

Suppose

pCx.

Therefore pEf(x)

Suppose derive

c~V x

(v):

Let

Let

~

s

IRgpI<~.

and

x=~ l~I

and

(i) :

Similarly

.

: iE~ > ,

Then

V ~ ~(P),

~

to both

~

by cHMTI]3.22. and

~.

We shall

Let the c o r r e s p o n d i n g and

sub-

g : ~ >- U, Note that

i.e.

base(~)~Dof

and ~ u ( r ) ~ W # O . Let p ' E e u ( r ) n ~ V

for some

p''eV=ee (p)

q'=g.q''

~,~e

and since

and

and therefore

for some

I{iCa

and

to

q''EW:~ g

(q)

and

is one-one.

: p'(i)*q'(i)}i~w,

s

V~Zd Sb~H Then

~w.

Let

By

and this

respectively V ~ U{eH (p)

~ e G w s c~ with unit

and

,

: perU}

are

Ibase(~)l =

and

an~

ext-

base(~')

W ~ U{eH (p)

fo~ some

J5

Let

~',~'eCs

(~U)EIs( ~', ~ ) .

U~Y=O

~

rl(ey)eis(~+,~).

by [HMT]0.2.10(ii)

By

such that

and

such that

and such that

Similarly, W.

~

there are

D' ~I rl v ~Ho( , ~)

~'=rl m rlD(~u)orl

Omp

respectively.

be a cardinal

respectively

and therefore

we have

to

~

U,Y

Then by [HMTI]7.25(ii),

rlD(au)eIs( ~ ,f/)

we have

sub-isomorphic

, ~

Let

be of bases

rl(eU)eIs(~+,~)

IHI=~

= H.

Cs

~+,~+eCs

Then

-isomorphic

~Gws~

base(V)Mbase(W):O.

rls

we have

Let

to

Then

some

such

Ay:Ax:Af(x) .

lUAxIp~q.

f : ~ >- U

IRgqi:i~i~w

~>IAUBUUOYI.

H ~ UUyU~.

~U~V

xEA

Let

p', q'E~U (r)

= I b a s e ( ~ +) I=~.

V.

by

IRgq'i~w

sub-isomorphic

unit

There is

q ~ (0

~ : ~ U (r)

Also,

, U~Y=O

= base(~')

Let

p'=fop''

IRgq' I~w

~

and

~(~W)

eu(r)N~V#O

IRgp'l<~

Assume

~

and

Then

Proof of

and

p ~ ~11d,

be induced by

q'6~u(r)ngW.

contradicts

and

is e x t - b a s e - i s o m o r p h i c

by

is regular.

I rlwEIS(~,~),

p(O)=q(O)

~(~V),

and base~5)~Dog. Then

IRgp'l<~

iUAylp~q.

is regular

~k~,

a contradiction.

therefore

)

s

qEf(x).

rl~(~V)EIs(~,~)

Since

by

since

base-isomorphisms let

I rlvEIS(s

Then

implies

W ~ aa(q)

be such that

By

qex

Proof of

~ , ~ 6 C r s reg. We show that

Crs

= : p~Y}.

~

with

since by

That is,

~)~ is

is sub-isomorphic ~

we have

to

181

3.1{.

VAW:O.

By

~

fore

there

~

this was

; By

~9

Cs

.

to

c~

proved

in the

to some

,

is

(i)

morphic

we h a v e

composition

Problem

Let

3.12.

exist

We

shall as

Definition on

I.

~ 9

need

exists

e

F,D,

reg

Let

Let an

to this

Shelah

Isomorphic

Then

for

to b o t h ~

6~

and

and the

an e x t - i s o m o r p h i s m .

and

to

let

both

6~

~

~

.

and

in f o r m u l a t i n g

Does

~

?

Problem

base

U.

Let

F

3.13

be an u l t r a f i l t e r

d ({q 9 =

ud F ~ Udc

: {i 9

Assume

Are

function

that

UdcF

d

such

E Is fIf,

if

Theorem.

counterpart 3 . 2 0 and

as a b o v e

base-isomorphism

c,

ultra-

and an

udAF~':eJL

.

See a l s o

F,D,c,d

there

and

f

such

between

A positive of

the K e i s l e r -

3.11. that

h =

U d c F "~eJ[

U d d D ~'~~ ? By

(i)

in the p r o o f

:

U d d D 9 Is~5 ?

(A,~ 9

be an a l g e b r a i c

there

f)[ ~ ~5.

function

is yes

some

.

: iEUF),~)-choice

and

would

Are

(ud dD)-lo foud A F

ext-iso-

paper.

with

~)[,~CCs

Ultrapowers

h9

and

: a9

answer

problem

~

sub-isomorphic

~9

and

Cs reg

of this

~ IU.

~_>~,

the

~

definition

(F,
answer

Let

~,~ 9

: iEOD),~)-choiee

that

to b o t h is a g a i n

following

: e•

There-

to b o t h

is

is e x t - i s o m o r p h i c

ext-base-isomorphic

are b a s e - i s o m o r p h i c

We k n o w

~

Let

: j<e> 9149

filters

(D,(base(~)

Then

s149

3.13.

=

there

IHI~,

3.12.

Problem

.. UddD'~(~

by

parts

:
B

Gws c ~

in s u b s e q u e n t

Let

ext-isomorphic

every

~.

the

H

base(~)=base(~):H.

[HMTI]7.17,

of e x t - i s o m o r p h i s m s 3.10.)

there

base

is e x t - i s o m o r p h i c

QED(Proposition

as w e l l

of

Therefore,

~

and

3.1Q(iv).

~ 9 Gws c~

since

,

with

as

proof

~

of

EHMTI]7.17

and by the p r o o f

of

EHMTI]7.13

and

182

3.14.

we h a v e Ws and

that

simple

when

to

from

than

a given

that

meeting

the

~. only

construction

line

we n o t e

Theorem base

hW F,p

~

for

3.14. and

~<~,

Let let

Let

F D {{AEI

(i)

Let

~<~.

Then

(ii)

Let

~<w.

Suppose

(iii)

Proof. : {FCI

Claim

: FCA}

~

~x

and

{p[Ax/s]

from

be

(p) .

let

: xEA,

sEV]

and

: p[Ax/q]Ex]

3.14.1.

Let

Let

:

x C Q >.

x C Q

and

x!gx.

(ii)

),( S b ~ , n gEHom(< Sb Q , ~ , Q N ,D EQ3 ij

in s u c h

complicated

construction

for

~

or to i m p r o v e

point

of

view.

Gws c ~ on

:

In

this

I

-unit

~ Sb

w

~.

with Define

x c W > .

below

be

h*[~WE

hold.

:

C s [ eg

and

h =

x _C W } . such

that

Let

A(X)V=O,

f { rlv~h. if

peV,

f*~

V#O, Then

=

Q d ~

(p)

and

~ N , D ! ~ ) > ). 1]

d

g = < {qE~

fE

is regular

x E A >.

i,jE~.

(i)

~

a construction

be a

is a s u b - i s o m o r p h i s m

the h y p o t h e s e s .

: pEF/q]Ex}EF}

:

5 <~(P).

f

clearly

find

(i)-(iii)

Then

V~

e.g.

is a s u b - i s o m o r p h i s m .

: p[F/q]Ex)CF] and

into

[HMTI]7.3Oa).

W

~~)

see

more

to

direct

is u s e f u l

to

The

ultrafilter

Then

EIsm(~,~V),

Assume

by

Let

an

Fel}.

W=~

eg.

above

a~,

hEIsm(~@W,

(p)

f = < {qEV

the

-s

to

much

problem

: p[F/q]Ex}eF}

:

: {FEI

Let

F

be

requirements

if

: {FEI

Assume

: < {qE~

open

be o r d i n a l s .

pEW.

~ h ~ < xU{qEe~W

x

I Cs

a,~

not

~ECs~

an

vague)

finite

wWsa~

is

Gws c~

we w a n t

and e v e r y

a rather

ext-isomorphic

then it

(somewhat

that

would

~EWs

for

above

~

Cs

construction

~<w

15

of

If

This

Cs

we g i v e

from

Gws~ ~

of a

to some

Below

csreg-s.

~E

structure

Cs reg.

sub-isomorphisms

-s i n t o

structure

of

works

of

concrete

the

that

below

Ws

is s u b - i s o m o r p h i c

to some

construction

concrete

a way

the

Gws c~

is s u b - i s o m o r p h i c

Cs - s and

the

every

:

3.1

4.

183

(iii)

If

~<~

g~QE

Proof.

gEIsm([~yQ, [ ~ )

(i). Let

xCQ

Let

all

AEZ.

Then

q6gxNgy

and

Thus

q~gx.

iff

(ii)

then

by

Ws

{reI

and iff

d e f i n i t i o n of

F

: pEF/q~3]EF

qEg(cix),

by

iff

f~a

and

i,jE~.

{FEI

be such that

Let

q6gx

qE ~

iff

~<w.

{FEI

: iEF

Then

and

: ier}EF.

Then

By

Next we prove

A(dd)gxlfCq.

(i)-

rlQ = g-1

By

( V F e I ) [ p [ F / q 3 E x ~ p[P/f3Ex].

we have

for

qE~xNg(Q~x).

{F6I

x=Qngx.

by

: (3bE~)pEF/q~DEx]EF

iff

~ d g~.~@Q.

imply

pEA/q3=qEx

iff

and

IFi<w

and

(iii). Suppose

~<~

Let

Then

is a filter,

: {i,j}cF}EF._

we have g

x,yESbQ

(3b6~)pEF/q3~Ex}EF

qegx,

= Ws reg

ZEF

: p[r/q~EQ~x}~F

-x c QNg-x = Q~gx

x _c Q,

Then

Let

geIsm(~&Q, ~ e ~ ) .

x c QNgx, Let

(ii)

{FEI

(3bE~){FcI

p[P/q]Ecix}EF

: FCA].

iff

q6cigx

: iEF

F d Do(D-q).

since

since

{FEI

Let

qEg(xny)

: p[F/q3Ex}EF

iff

qEx.

Z d {AEI

EQ] ) = D 13'' (~) g(Dij

iff

is a s u b - i s o m o r p h i s m and

Cs reg.

q6Q = ~ ( P )

{FCI

then

since

~SECs reg.

Agx = Ax

and

Then by the

fEgx.

Q E D ( C l a i m 3.14.1.)

Proof of

(i) :

geIsm( ~ Q ~ , ~ ) Z d a~W. 6.2,

Let .

Then

k d < xUfx

x<~.

Then

Let

Proof of = g(x)NQ =hx.

A(~)Z=0,

hence

: xCW )EHom(/~

(ii):

Let

Now 3.14.~(iii)

fore

q6V. F d

A[WIQ=Q.

Let

~)

Observing

k = hw F,p

"

completes

(i).

(iii) :

~<~.

Let Do(D~S )

(VA~Z)pEA/s]=s.

Suppose

hence

W = Q.

~][c [@Q, V c ~

First we show that

s d pEAx/q~.

Then

is finite 9 Let By

AxlsCq

Let

gx = x ~ { q ~ Q

completes the proof of

Let

heses of 3.14(iii). Let

since

we have

f d r l Z o g o r l Q E H O m ( [ ~ W ' ~[Z). By [HMTI3

'

by 3.14.1(iii),

Proof of

By 3.14.1(iii)

gorlQeHom(~& W, ~ )

--

the proof of

~ d ~W.

xCQ.

: {P~I

Then

: p[P/q~Ex]~F}=

(ii).

and

f

be as in the hypot-

fx : V ~ h x = {qeV

s~Q Z d {A~I

x =

: pEAx/q3~x}.

by the h y p o t h e s e s and there: FCA}.

and r e g u l a r i t y of

f~

Then

Z6F

we have

and pEA/s]~x

~

3.15.

iff

p[A/s3Ex

iff

{AEI

for every

: F~A

and

AEI.

Now,

p[A/s~ex}EF

fx={qEV

: p[Ax/q3Ex].

and by

A(~)V=O

the cylindrifications. iff

iff

Next we show

it is enough

p[Acix/q]Ccix

Let

q6V

Let

F ~ {iEAx

sEQ.

Thus

q' ~ q[F/s]EV

p[Ax/q']. then

I.e.

Qcv

and thus

morphism. and

It remains

qEV

and therefore

QED(Theorem

3.14)

3.15b)

For the necessity

Corollary a)

Every

is

Ws n L f

~eWs

morphic

Remark

iff

qEcifx,

by

Then

Now

C~)V=V.

IFI<~

p[Ax/q']Ex

by

that

f

is regular. Afx=Ax

by

Axls c If

pEV

is a sub-isoLet

xEA,

we have

s~fx = {g~V

Let

pCAx/q]EQ,

fCIsm(~,@~V).

below is a g e n e r a l i z a t i o n

and every

IAxNHI<w

:

iff

By

of the conditions

Ws n L f

to some Let

(~bC~)p[{i}UAx/q3~Ex

s~fx

p[Ax/s3

=

: p[Ax/g3~x}.

of CHMTI33.22

and 7.27.

in a) and b) see [HMTI]7.3Oa)b).

3.15.

base;

b)

iff

f{:~

by

is a h o m o m o r p h i s m w.r.t. iff

which implies

q~fx

By 3.14.1(ii)

qEf(cix)

A(~)V:O.

Afxls < q.

: p[A/q]ex}EF We have seen

qEV. Then

: qi~si }.

to show that

= p[Ax/q3

f

We have seen that

fxDx

be such that

Corollary

by

q'efx.

s=p[Ax/q3Ex.

and

pC{i}UAx/q]Ecix (3bE ~ )q ~ fx

{AEI

feHom(~, ~V).

ice, xEA

iff

SExEA.

iff

to show that

(3b6 ~ )p[Ax/q~l~x and

qEhx

is dropped, hF~HOm(~V,

.

Thus

have unit

some

pCAX/S3EX}

3.16.

csregnLf

and let to

sub-isomorphic

:

au(P)

Cs r e g

] = I csreg~Lf

Let

HC~

be such that with

3.14(ii)

namely for any ~ h F (V)) 9

N(Ws N L f

xeA > E I s ( ~

(i) Thm.

some

with

the

same

with nonempty base is ext-isomorphic

qEaU

~,~EWs

to

~Aw~

unit

aU ( q )

.

be such that H]p~q.

Then

Moreover

~

(VxeA) is iso-

( {sEaU (q)

, ~).

becomes and

false if the condition p~,

V ~ ~x (p) we have

~<w

:

3.17

185

.

(ii) E

By P r o p . 5 . 6 ( i )

Zd Sb e~

such

(ii)-(iii)

Remark

that

do not

Let

extend

does to

w~:I~l~. Ws

such

(i)

u Cs A H ~ :

(ii)

(V~EGws~OmPNH~)

(iii)

If

(iv)

IA]+IBI~Iel .

not

for all

1<~<~

there

preserve

regularity.

reg

Ws

(GwsC~

that

LC~.

from

Then

(i)-(iv)

for all

is

Hence

V Thm.3.14

.

(Vq~l ~

then

there

below

are

fie Ws n D c

and

hold.

u~. ) Ibase(~)~q~;m]~.

~ ~ D c e.

is an e a s y m o d i f i c a t i o n

is to r e p l a c e where

Let

oCs

[~LI~w

The p r o o f

"max{wqu

of

: O
[HMTI]7.3Oa).

with

The

"max{wqu

basic

: O
change every-

in t h e proof.

(2)

If we

obtain

replace

statement

(~'~)

There and

(*)

is

Then

[HMTI37.3Oa).

Ws N H ~

choosing

it.

is true

laI:~,

for

case

.

and

in

[HMTI]7.3Oa)(4)

such

of

~w +

KE{Ws Then

~EWs

that

t h e n we

for all

of

, G w s ~ Omp}

(~)

NDc

qE1 ~

the p r o o f

~GKNH

,

is true with

with

by

iff

IAI~a.

of

~ w +.

In

Then

there

Rgq=base(~.5).

[HMTI33.18

is i m m e d i a t e

by the p r o o f

and u s i n g

The Ws C I C s reg.

3.17(1)(ii)-(iii)

above,

I~L]=m.

4.

About

"K"

IAI~

and

iterating

The

with

K = Ws NDc

are

We o m i t

by

Ibase(~)~Rgql~.

(*)

let

by

K

,

particular:

goes

"

below.

Let

~E

"Ws

~E

@1 ~

~ .

proof

hF

that

3.17.

(i)

Let

we h a v e

Propositions

4.1-4.3

Sub a%~eb r ~

below

see

[HMTI34.2

statements

(i)-(4).

186

4.1.

P r o p o s i t i o n 4.1.

Let

s

be a full

Gws

.

Then

(i) and

(ii) b e l o w

are equivalent. (i)

~

is normal.

(ii)

Every s u b a l g e b r a of dimensional

Proof.

(i) ~

Suppose

~

a regular

Let

{kEy p~Ex

xEC

and

and

by

Then

p~EV~x.

bEW

~w,

and

d e f i n i t i o n of subunits p~ECoX V,

since

p~Ex.

Thus

QED(Proposition

CoX

~

x

~

~

be V.

is not regular.

Define

and

since

a~W

A(CoX)=O ,

and

Ax=l

Y~-W#0. x since

Thus

iUA(CoX )

by EHMT]I.6.8.

A[V](~w(q))=O q~@CoX. q~

by the

At the same time

and

p Ob

but one of them is in

CoX

are both in while the

is not regular.

4.1.)

P r o p o s i t i o n 4.2 b e l o w implies that there is a s u b a l g e b r a of w h i c h is not regular. "full"

and

is regular.

(see Def.O.l).

CoX

Since

there are two

YAW#O

is full,

(i).

We shall exhibit

furthermore,

such that

since

(ii) =

is not regular.

Now the two sequences

they c o i n c i d e on

other is not.

is not normal.

Let the unit of

Clearly, CoX

~

and

of

x~C

O qa~V

1.3. Next we prove

by EHMTI]I.6;

aEY~W.

: k(O)=a}.

and

Ax:l

~W (q)

Now we show that q~EV

Gws

such that

~Y(P)

and

is regular.

(ii) follows from Thm

is a full

b6yew

g e n e r a t e d by a set of locally finite

regular elements

is not normal we have subunits

~

Gws

the m i n i m a l

It also implies that the c o n d i t i o n

is n e c e s s a r y in Prop.4.1.

Proposition

4.2.

Let

~eGws

(i)

~]l/&(~)

(ii)

(VY,%~Subb(f~))Eynw#o ~

(iii) c(x)d(~•

.

Then

(i)-(iii)

below are equivalent.

is regular.

is regular

IYIN~=IWIN~]. (in 0t) for every

~<~N~.

4 . 3.

187

Proof. unit

We may suppose V.

Let

Subu(V)

aAu = {~y

by [HMTI31.17, ~.6.

Let

4/EGws

(pi)

~<w.

Let

: iEI}.

Let

with

a

1

c(~)d(•215

Then

a}IEMn(~ )

A(a )=0

and

a• = u{~Y (pi)

: iEI

i

First we prove Let

kEa•

IYilA~

qEV

by

(ii) ~

and

kea~.

Therefore

qEa

Then

,

i.e.

and

Let

r~EV~a

Next we prove a EMn(~). ~<w. Let

By

~y(r)

and

and

(i).

(iii)

(i) ~

: ~<~,

X

is

is regular,

by 1.3.5(i),

d..

i,jee}.

H-regular,

k ~ Y i(pi) ,

of

y ny1 n#O

V

s~ea

a

Then

by

(ii)

by

k(O):q(O). fails. Then

such that

Then

~<~.

ynw#o

is not regular.

a

by 1.3.4(i).

(V~
is regular for every

is regular, H { 1.

and

IWI~, b~W

(iii) holds t r i v i a l l y by

Let

1.3.4(i).

Let

q~y(p nn)

Suppose that

suppose that

we have that

X { {a ,dij

~W (s)

Therefore

i.e.

(ii) holds.

(ii) since

• { IYI+I.

(iii) ~=

0t~Gws

by

is regular.

iYI<~.

element of

:

a

bCY,

Suppose

Suppose

IYnIh~

bCYnW by

Suppose that

k(O)=q(O).

there are two subunits iWi>IY]<w.

(iii).

'

for every

Then

xCDm H_

Therefore

By EHMT32.2.24

i,js and every

~(~i

we have

6[)X Mn(~)

:

Sg ( ~ ~) x.

QED(Proposition

4.2.)

P r o p o s i t i o n 4.3. (i)

Let

~.

Then

The g r e a t e s t regular in general. of

~

Namely:

s u b u n i v e r s e of a

There are an

such that

Ax=Ay=1

regular subalgebras (ii)

Lf

(i)-(ii) below hold.

in

The g r e a t e s t regular

~ Lf

~EGws

and both ,

but

Gws

need not exist

and elements {x}

{x,y}

s u b a l g e b r a of a

and

{y}

generate

does not. Gws

may exist even

if regular elements do not g e n e r a t e regular ones. Namely: are an CoX

0iEGws

and

is not regular and

algebra of

xEA

such that

T~(~)

x,y

Ax=1,

x

There

is regular,

is the g r e a t e s t regular sub-

188

4.4.

Proof.

Let

~w,

~(P)u~(~+a) P r o o f of

p ~ (0

(r)

s {

(i) :

: (q0 E~ = q0

Let

is odd)

Sg{y}

are r e g u l a r .

since

Ax=Ay=l.

algebra, =

(r)

P r o o f of Ax=l,

x

that

Ts

regular that

which (ii):

generates

: qo>~

x ~ {qeV

and

that

y =

( ~ x i<mj ~(i,j)

: i,j~},

,

y

-dij

generates

occurring in

Y,

let

and

and

[HMT32.2.24

in any r e g u l a r and thus

of

sub-

c0(x'y)

Now

~ ~(s

~

yeA,

We show

9~I~(~) l~Mn(~).

By [ H M T 3 2 . 2 . 2 4 ,

x i d: {qEV

-x

~

is not r e g u l a r .

Let

element.

: qi~}

is

We show

A = Then

yEA

d j) ~(i,j) and

H i

: i,je~}.

such a

i.e.

and

subalgebra

where

~ i
1.3.5(i)

Sg{x}

:

(i) is proved. : qo~}

n,mj,nj,u(i,j),v(i,j)~,

{6 i : i
V

y ~ {q6V

Now

and qo is even}

I~+~I:I~I 9

an i r r e g u l a r

and

is even)}.

CoX = ~ ( a + ~ ) ( r )

since

implies

and let

is not c o n t a i n e d

is the o n l y r e g u l a r

for some

is even}

(q0~e = qo

{x,y}

Let

~){xi,dij

Then

: qo

is not r e g u l a r .

= Sg ( ~

since

and

xny : {qCV

by P r o p . 4 . 2 ,

~ j
x ~ {q6V

T h i s c a n be seen by u s i n g

is r e g u l a r

y

r ~ ( ~ : ~<~ >

[@V

However,

since

(~+~)

: ~<~ > ,

y'.

Let

H

dj =

denote

H ~ {u(i,j),v(i,j)

6. w h e r e z (Vj
the set of i n d i c e s

: j
i<(mjUnj)}oA(d).

since

y c _ ~(~+~)(r)

IHl<m.

C_aase i

(Vj
and

y#O.

Case

2

(3j
= ~(a+~)(r),

sin c e

Thus

QED(Proposition

Remark

4.4.

(ii)

C ( H ) y = ~ ( ~ + ~ ) (r) ,

Then

we m a y s u p p o s e

O#(-y'd)

Thus, in b o t h cases, element.

Then

y

nj#O.

Then

C(H)(-y'd)

c ~(~+~)(r)

generates

~ ( ~ + ~ )(r)

which

is a n o n - r e g u l a r

is p r o v e d .

4.3.)

By P r o p . 4 . 1

=

w e h a v e that

locally

finite dimensional

:

189

4.4 .i.

regular

elements

generate

If we do not suppose can destroy

regular

normality

regularity

See

1.3.5 w h i c h

locally

always

generate

negation

(apart

dimensional

that

regular

ones

> i,jE~

of


from 0,i)

operations

Proposition base

~

in fact

in every Gws

finite

normal

-s then

finite

in the

.

cylindrifications

dimensional

the c o u n t e r e x a m p l e s

Gws

elements,

in 4.1-4.3

dimensional

and

regular

preserves

. (Note that

regularity

in

reduct"

Crs

of locally

see

elements

"cylindrifications-free

~EGws

but

-s only

finite

elements.)

If we do not require Boolean

of

of locally

only c y l i n d r i f i c a t i o n s . : implies

ones

locally

can d e s t r o y

4.4.1.

Let

there

are d i s j o i n t

dimensionality

regularity,

~

such that regular

finite

and

~>i.

already

There

elements

generate

regular

x,yeA

then

in

is

Cs

.:

~ECs

nonregular

such that

again the

nDc ones

of in

~ [ ~,

xUy = x 9 y

is

not regular.

Proof. Set

Let

a~e

R ~ {qE~

d y = RNX. Ax=H,

~>i.

: {iCH

Let

Ay=H

and

~

: qi~O}

be the

and hence

is not r e g u l a r

since

x

4.4.1.)

To c o n s t r u c t

regular

the q u e s t i o n

regular

elements

y

base

R@{I ~

generate

Let

V

is d e f i n e d

to be

Q-weakly we have

~

small

x ~ {q6R generated However

frequently related

(not n e c e s s a r i l y

be a

JHI~I~HIA~. : (ViEH)qi=O ]. by

{x,y].

x~y : xUy = R

,Oj.

we shall

regular

that

and

are regular.

and

"which

4.5.

KcACV]x

be such

They are closely

Definition

infinite

with

algebras

4.7 and 4.9 below.

address

and

HC~

is finite}

Cs

AR=O

QED(Proposition

4.6,

Let

need P r o p o s i t i o n s

to Thm.l.3

finite

and they

dimensional)

ones".

Gws

-unit

and let

(O-wsmall)

in

xCV,_ QCV._ V

Then

iff for every

x

190

4.6.

(VF -~c ~)(VqEQ)(]@ c_w K) c(0){q} -~ c(r)x. x

is said to be w e a k l y small

(wsmall)

Note that w e a k l y smallness is since

x

is small in

(VF -c~

~ ) ( 3 e -c~

Theorem

V

K)(Vqev)

in

V

if

x

V-wsmall.

a w e a k e r p r o p e r t y than smallness,

iff for every infinite

{q}

c(e)

is

~ c(

-

F)

x

KcAEV3x

"

1.3 says that small regular elements g e n e r a t e regular ones

in normal

Gws -s. The next P r o p o s i t i o n

4.6 says that w e a k l y small

regular elements generate regular ones in normal g e n e r a t o r elements)

P r o p o s i t i o n 4.6.

Gws -s, if they

Let

f][EGws n~

(VF --~0c~)C(F)xnc(F)Y:O) .

be g e n e r a t e d by a set

Assume that

Then

~

Proposition G~A.

Let

Let

QC_V, A[V~Q= O

wsmall. A s s u m e c o n d i t i o n s

U[EGws n~

is regular.

(i)-(ii) below,

(VxEGN{ yJ ) C(F) xnc(F)Y:O.

(ii)

(VfEY) (3q6Q) (Vp)
~

II)

rlQ

(1)-(III)

w i t h unit

iff

for every

Recall the

be g e n e r a t e d by

y~G

G and

is

Q-

F c

~.

q[F/p]Ey3.

b e l o w hold.

is regular if every element of is an isomorphism,

if

~(~%)

G

is regular. is simple,

EG) IAyIA~. III)

V

and suppose that every element of

(i)

I)

of

from def.l.3.i.

4.7.

Then statements

G

(Vx,yEG) (x~y ~

P r o p o s i t i o n 4.6 is a special case of the next Prop.4.7. Dm H

(the

are "very disjoint".

w e a k l y small regular elements.

notation

we have

(VH~a)DmHNIg(i]L) (G~DmH)={O}.

To prove this proposition,

we need two lemmas.

and if

(rye

4.7 .i.

Lemma Let

191

4.7.1.

Let

Q~y!1 ~

~A~

satisfy

and

conditions

every

zEA,

statements

(I)

(HF --~c ~ ) z n c ( F ) Y ~ O

(II)

There

is

~Gws n~

let

0 -c- W

(I)

and

~

~

(i) (II)

4.7.1.i,

Lemma

such

4.7.1,

4.7.1.2

below

(ii)

below

generated

by

GC_A.

Then

of P r o p . 4 . 7 .

for

hold.

(3F --~c ~ ) z n c ( F ) Q ~ O . that

(VfEy)(3qeQ)(Vp)[fEF/p]Cz

To prove

and

be

we

every

iff

need

will

for

be

have

qEF/p]ez].

in t u r n

used

we

0 c F C ---W

in

a definition

subsequent

and

parts

two

of

lemmas.

this

paper

too.

Definition

4.7.1.1.

Let

r b p ~ ( (( f p i , ( ~ R g p ) I f

P

>

be one-one.

: B >-

: ic8 >

f

:

Define

is a f u n c t i o n

and

Rgp~Dof

rd P ~ rb p* "

Lemma

4.7.1.2.

Then

(i)

and

Let (ii)

(i)

rd P E

(ii)

If

below

Is(~

V

p

: B >~ hold

p~V,

is a

~

and

be one-one.

Let

rdPV

is

CrsB-unit.

and

rb p

~rdPV)

Gws

-unit

with

a

V

: V >~

subunits

be

a

rdPV,

{~y(pi).

Crs

if

: iEI}

-unit.

8#0. then

1

rdPV

is a

Gws

-unit

with

subunits

B {B(yi•

Proof.

Let

-unit.

Then :

g

Then

P

Notation:

Proof and

: i~I,

of

y(q)

d

(i) : {gop

: B >-- ~ For Let

be

every qEV

: gEV(q)} "

f(q)EHo(~P~V(q), and

Rgf(q)

b(q)

defines

a base-isomorphism

geV be

f(q) by

Let function b(q),

g,

d V(q)

d < {gop

: gEx }

d

on

base(Y(q))

see

EHMTI]3.1.

d {gEV

since

< (u,q')

V

be

a

Crs

(~NH) Ig.

Define

EHMTI]8.1.,

b(q)

Let

H = Rgp.

denote

fixed.

Let

d

Let

we

~Y(q))

= SbY(q).

is a o n e - o n e

one-one.

g e ~ Y i(pi)} .

:

: g, _c q} xESbV(q)

( V g C V ( q ) ) (gop) + :

: uEbase(Y(q))> and

) .

therefore

9 b(q)

-

192

4.7.1.3.

Let

= "b(q)(Y(q)) d -~

W(q)

~}W(q))

and

: b(q){gop every

W(q)=W(g)

~>i.

Let

iff

if

~W(q))

q'=g'

and by

we obtain that and

Proof of

~ ~P[@v,

rb P

(ii):

rdPeHo(~,~),

is one-one Let

{~y!pi)

on

to

: q E H / P i ] E ~ y i(pi)}.

and therefore

Then

therefore Y(q)

for some

is one-one

: iEI} = Subu(V) V(q)

= {q[H/g]

: ieJ}.

Let

~y!pi)l : ~y~pk)

is a

Then

Let

. By

6~V(q),

because

qEV

W

Let

J ~

: (~iEJ)g E Hy!H1pi)}1 { By!piop)

for

iEJ.

l

q[H/g~

which implies

Gws -unit with subbases

~

"

1

i,kEJ.

g~EV(q)

and

"

Y(q) = o{BY! pi~

gEwiAw k

~

by

V.

1

Let

then

SbV(q)IrdPEHo(~P

1

d= {iEI

iff

RggNRgg~#O

since

rd p

Let

A(~)W(q)=O

iEH, g C V ( q ) , g ~ e V

(VqEV)

~W.

V(q)=V(g)

Therefore

[HMTI]6.2

for

~{

iff

g~eW(q)

Therefore we may apply

EHMT]0.3.6(iii)

rd p = rb P'"

then

=

: gEx} = rdPx

V(q)mV(g)#O

since if

h(q)EHo (~f6~V(q),

h(q)x = b(q)(f(q)x)

: qEV},

iff

i gaEW

A(~)V(q)=0

Now

: gEx} = {rbP(g)

W ~ o{W(q)

gEW(q),

g, = (gai ) ,

[HMTI~6.2,

since

base(W(q))Nbase(W(g))#O.

iE6,

Also,

d : ~b(q) of(q).

= SbV(q)Ird p

Then clearly

because

h(q)

: gEx} : {b(q)~gop

x ! V(q).

g,qEV.

by

h(q)

and

p k ) l

Wi=W k.

{By!piop)

and

This shows that

: iEJ].

This

immed-

1

iately yields Lemma 4.7.1.2(ii). QED(Lemma

Let ~U

4.7.1.2.)

V~U,

f,qEV

as follows.

t(f,q,H)(s)

{

and Let

H~.

sEV.

EHIs3

Define

the function

t(f,q,H)

: V

Then

if

( ~ H ) lq!s

if

(~--H)IfCs.

otherwise

Lemma

4.7.1.3.

Let

V

that the bases of the coincide. E Is(~P~v,

Let

H c

~P~v)

(unique) ~

.

be a

and

Gws -unit,

~.

subunits of p : IHI > ~

V H.

Let

f,qEV

containing Then

be such f

and

t(f,q,H)* 6

q

4.7

. 1.

193

Proof~

Let

= {~y!pi) 1

Let

V,f,q,H

: iEI}.

W ~ rdPV.

Then

b

: base(W) and

(ii)

and

by

Lemma

4.7.1.2(i)

IHI<~.

show

have

t(f,q,H)g

g'

= q'

QED(Lemma

N o w we with

turn

V, A : Sg G

since

follows

4.7.1.

that

and

let

of

containing

Lemma iff since

if

u,f'>

if w=< u,q> .

by

By

iff 4.7.1.2

EHMTI]3.1.

By

Therefore

rd p = rb pe

for e v e r y

rbP-1(borbPg)

otherwise

by L e m m a

V, ~ W ) .

obord p.

w

w=< u , f )

gcV.

it

it is

Let

geV

= rbP-1(bo<< gpi,g') Suppose

= rbP-l(bo<< gpi,f'> = t(f,q,H)g.

: g"

Let 4.7.1.3.

t(f,q,F){:y=y

@

Let

that

IF[ > - ~

F.

iff

@ c --

= O

coincide

of

The

to p r o v e

case

q,

iff

V

c(r)ync(F)x

for

E

of

subset

the

is(~P[@v, that

= O.

xEG~{y} Let

of

Let Then

subunits

and

F#O,

~P@~v)

by

t(f,q,F)*x=x

G1t(f,q,F)* if

z,

qEF/p]~y].

is n o r m a l

we h a v e

and

(II)

be a r b i t r a r y .

Therefore of

n~

conditions

nonempty

the b a s e s

t(f,q,F)*

{~Gws

satisfy

~

--W

since

t(f,q,F)

choice by

F c

Then

qEF/p]Ex].

by the

ocvcV

(Vp)[fEF/p]Ey

Then

~w,

be a f i n i t e

qEF/q]:q6OCv. q

Let

It is e n o u g h

@ ~)G.

and

q[F/p])nx

4.7.1.

that

Let

By d e f i n i t i o n

(Vp) E f E F / p ] E x

(Vp){fEF/p],

(II).

since

p :

assume z~A.

be such

f

of L e m m a

~

c Id then be

=

:

analogous.

Let

from

q~Q

e y c V

e--F#@.

p ~

rbP-1(borbPg)

and

z E Sg(~

fE~/q] V

is(~

u,q'>

f[H/p]EV

ge~y!} pi) l

: iEIHI> ) = q E H / g ]

to the p r o o f

of

f~y

since

Is(6~W,~W)

rd p E

~

~

=

g':(~H)Ig

: iEIHI> ) = g = t ( f , q , H ) g .

and

is e n t i r e l y

(i),(ii)

such

~ 6

b(w)

Subu(V)

4.7.1.3.)

unit

(I)

: ieI,

Let

we d e n o t e

: W

= rbP-1(b~

: iCIHI) ) = rbP-l( << g p i , q '> of

b(W)

Then

g : f[H/g]

g~V

define

t(f,q,H) e = rd ~-I

g'9{f',q'}-

Then

every

and

that

: I ) = rbP-l(<( g p i , g '> "~ = f'

in the h y p o t h e s e s .

w~base(W)

Therefore

we

enough

that

For

u{IHI (Yix{g'})

to s h o w

be s u c h

every

be as

> - - base(W)

W

is e n o u g h to

p

Notation:

For

qEH/p]EV

and

the

194

4.7.2.

subalgebra

of

zER

z E Sg ( ~ p

by

iff

~P

@IV

generated

~)G.

Then

by

G.

Then

R1t(f,q,F)~{~Id

t(f,q,F)~;z:z

i.e.

and

(Vp) EfEF/p]Ez

qEF/p]ez].

QED(Lemma

Lemma

4.7.1.)

4.7.2.

Let

EGws n~

Let

Q~V,

A[V3Q=O.

a.

(VzEIg ( ~ )G,~{O})zNQ#O.

b.

For every

with unit

Assume conditions

H~

and

V

be generated

by

G~A.

a. and b. below.

zEIg ( ~ ) (G~DmH)

we have that

(~qEznQ) (V~ --we ~) (~@ --~c ~N(HU~))C(@){q}~z. Then statements

(I)-(III)

below hold.

(I)

~ is regular

(II)

rlQEIS~,

(III)

(~H~e) Ig ( ~ ) (G~DmH)NDmH:{0].

Proof.

if every element

if

~(~)

xEAN{0}. Ease i

is regular.

is simple and

(II) :

Then

Suppose

x:d@z

zEMn(~).

T/~(~)

for some

Then

(~yCG) I A y I ~ .

= rlQ(X)

# 0

Case 2

by

EMn(~)~{O} E d@z= x

dCMn(~)

and

If

d=o

Let

Suppose

qEz.

Ed

since

and

qEQ

and

c(@)Q:Q

a.

@hAd=0. we have

(II)

is proved.

(III):

Let

Let

Let

by EHMTI]5.1.

we have

rlo E

Therefore

xNQ =

q6xNQ.

x=zEIg ( ~ ) G ~ { 0 } .

q~dNQ.

(30 ~we~Ad)c(@){q}

condition

A[V]Q=O

Suppose d#0.

b. we have that

Proof of

zEIg ( ~)G,

is simple.

then

by Case 1.

and we are done.

xnQ#O.

By

~(~)

and we are done.

q6d

and

(VyEG) IAylZw.

xEMn(~)--{0}.

z~Mn(4J[). xnQ~O

is simple and

xEMn(~).

Ho(~(~)[)) .. We have that

that

G

Assume the hypotheses.

Proof of

then

of

~ z.

Therefore qEO/p3EQ.

H~e. Let

Let ~ c

Then If

Then Let

By condition

dnQ#O

qqz

then

qEznQ

by

de qEd-z

and by condition

q[@/p] q z. Now

qE@/p]~

qEO/p]Ed- z E d@z = x. Therefore

qL0/p]ExqQ

By showing

x E Ig ( ~ ) (G~DmH) , x#O. Then ~.

Then

a.

xOQ#O by

4.7.2 .i.

(~@ -c~

195

~ ( H O g ) ) c (e) {q} ~ x,

qEx,

and thus

IAx~HI~w,

Ax ~ Hu~. x ~ Dm H.

Therefore

@nAx ~ 0

by

Let

conditions

~ECA

(ii)

Dm~ = Sg ( ~ ) (GNDm~) .

(iii)

Sg(G~H-dim)

c H-dim,

(i) ~ (ii) and s Let

By

Fact(~{)

By

G~DmH~I

was arbitrary,

this

shows

(III).

be g e n e r a t e d

(i)-(iii)

D4NIg(~)(G~Dm~)={0}.

Proof of

~

--W

This proves

(~)

Proof.

~ c

(I) we need a lemma:

Lemma 4.7.2.i. Consider

Since

--

i.e.

To prove

Then

by condition b.

-

where

and let

(i) = (ii)!:

and let

H-dim { Mn( ~ ) U { x E A

HC~.

: IAx@HI<~}.

~ (VHESb~) (iii) holds.

Hc~.

Let

I { Ig(~]L) (G~DmH).

in the proof of 1.3.3 we have

GCA

below.

(i) holds]

~ECA

by

G~I~Dm H

Assume

we then have

and then by

mmHnI={0}.

DmH!Sg(~)

DmHESU~

(G~I).

we have

Dm H = Sg(GNDmH). Proof of

(i) ~ (iii~:

D m~nH • (~ ) ( G ~ D m ~ ) = { O } . ~(~)G'.

Let

G'

and

G

~'

: IAy~Hl~}

~ I g ( ~ ) (G~DmH).

(G~Dm H) = {0}

and

satisfy

be arbitrary

G'NDm H' = {yEG'

Ig ( ~ ' ) (G'~DmH')

DmHNIg(~)

~J[

G'~G

First we show that

mm H' { mmH((]~'). then

Suppose

then we have

By

(i), i.e. 6[' d=

and let

satisfy ~ G~mm H,

(i), too. and by

Dm H' 5 mm H

Let

~' [

and by

DmH'NIg(~')(G'~DmH')

= {0},

as desired. Now we turn to the proof of be arbitrary xeMn(~). (i) For,

Suppose

K ~ Ax. IH 9 K I ~ .

Let

G' { GnH-dim.

We show that

IH 9 K I ~

Let

xESg G'

implies

Now

G'NDm K ~ M n ( ~ ) . note that

and so then

and let

(iii).

KNH

H-dim ~ DmH,

and hence

is finite and hence

[AyNKI<w,

so

IHNAyl

HNK

Sg(H-dim)

~ Dm H.

is infinite.

is infinite.

Hence

If

y6Mn(~J~)

So

xEDm H

yEG'ADm K by

196

4 . 7 .

yEG'~H-dim. we h a v e

Thus

(i) holds.

G'

and

~'

= Sg ( ~') (G'r]Dm K ( ~ , ) )

that

DmK({~')

By

xeDmK(~')

(ii).

Since

then

xCMn(~').

= [~G'

satisfy

_c M n ( ~ ' )

We have

by

,

seen

(i),

(i) ~

Sg(G~H-dim)

C H-dim. QED(Lemma

4.7.2.1.)

N o w we can p r o v e that

( V HHC ~ )_ D m

Lemma

1.3.6 w h i c h

(I) of 4.7.2.

= Sg(GnDmH). yields

By

(III)

T h e n by

that

~

and by L e m m a 4.7.2.1 We have

~ E G w s n~

is r e g u l a r

we can a p p l y

if e v e r y

element

of

G

is regular. QEm(Lemma

4.7.2.)

N o w we turn to the p r o o f of P r o p . 4 . 7 . e r a t e d by

GCA.

(ii) of 4.7 Let

zEIg ( ~ ) G ,

4.7.1

of L e m m a

F --~C e

and

)y

for some

satisfied

by

Bv~

(i) of 4.7 and a s s u m e

F --wE a Qny

AEV~Q=O

yEG.

y. T h e r e f o r e

we have

Let

By yqDm H

since

'

qEznQ

we h a v e

qEznQ

then

we h a v e y

is

that e v e r y

and

Q =

that

Q-wsmall.

Then

by

We h a v e seen t h a t c o n d i t i o n

4.7.2 y i e l d s 4.7..)

the c o n c l u s i o n

of 4.7.

: yEy}

for some

for some

y6y.

(30 c

By c o n d i t i o n

since

qEc(F)y.

z _c ~ { c ( F ) y

of

is s a t i s f i e d .

Z ~ C ( F ) L~ Y

qEc(F)ynQ

IKI~w.

element

~ --~c

c(0){ q} ~ z,

QEq(Proposition

and

and

b. of 4.7.2

satisfied. Hence

(i)-

znQ#O.

By F

be gen-

(VzeIg ( ~ ) G ~ { O ) ) z n Q # O .

(3 F --~c ~)znc(F) -Qr

Y ~w G~DmH"

c K)c( {q} ~ c( --w O) -

C(@uF)Y

are

zeig ( ~ ) (GNDm H)

K ~ Ay~(HU~)

t h e n have

i.e.

znc(F)Y~O

zEIg (/j[) (G~DmH) . By

gJ~EGws nOrm

F i r s t we s h o w that c o n d i t i o n s

We s h o w t h a t t h e n c o n d i t i o n

HC~_ and

be a r b i t r a r y .

.

holds,

4.7.1

now condition

Q-wsmall.

Let

Then

therefore

Assume

4.7.2.a

z~o.

(I) says t h a t

= c(F)Q,

Let

Q ~ Z d @@I ~

imply that

The c o n d i t i o n s

is

Let

Let

and

(i) we

c(0){q}

b. of 4.7.2

is

G

4.8.

197

Definition (i)

4.8.

Let

qEV,

Let

V

be a

Gws

F C

~

and

KCe.

(q,F,K)-small

-unit

x

is d e f i n e d

(VqEx)(VF

Note

that

Proposition eliminated

c

i-smallness

Every

is r e g u l a r .

Proof.

We may

4.9 w e

need

Definition (*)-small

to b e

is a s t r o n g e r that

if w e

the

normal

by

a definition

and

~ x

(i-small)

is

property

iff

of d i s j o i n t n e s s

generated

[HMTI~I.17.

V

than wsmallness.

from wsmall

Gws

in

(q,F,K)-small~.

condition

change

~kw

Let

h q c(S)x.

can be

to i - s m a l l .

by i - s m a l l

Before

giving

regular

the

proof

a claim.

~Gws

and

x~A.

~ x

is

Tnen

x

is

said

to be

iff

(VqEl ~ )(VF

Note

assume

4.9.1.

xCV.

is d e f i n e d

irreversibly-small

shows

from Prop.4.6

4.9.

x

--coc c~|

~)(VKcA[V]x)EIK[_>~

elements

of

to be

4.9 b e l o w

Proposition

Then

let

iff

(~@ --w C c ~ r ) (3hEC [(Kn@) v] {q})(v~ (ii)

and

that

c

--W

~) ( V K C A ( ~ ) X ) E I K I > _ w

(~)-smallness

appears

t o be

(q,F,K)-small3.

a stronger

property

than

i-

smallness.

Claim

4.9.2.

x

(*)-small.

is

Let

~EGws

Proof.

It is e n o u g h

Suppose

x

to p r o v e

is i - s m a l l .

We have

to p r o v e

then we

are done.

that

and

Let x

Assume

is

let

that

qCl ~

xEA.

Then

i-smallness

,

r c

~

(q,F,K)-small.

q E C(A){p }

for

some

is i - s m a l l

implies

and If

x

KCAx,

iff

(*)-smallness. IK]~a.

(VA --~c ~ ) q q c ( A ) x A --we ~

and

p6x.

198

4.9.

Then

x

(3@ c

is

(p,FUh,K)-small

~N(FUA)) ( k~C(K~@)

C(K~@){ q}

such

and t h e r e f o r e

{p))(V~

that

c ~@)k --~

k 6 c(A){h)

~ c (~)x.

since

It is e n o u g h

--~c ~ N @ ) h ~ c(~)x.

Then

C

~._@

that

by

x

@ c ~NA.

is

QED(Claim

~ --~c ~ @ . Then

h @ c(~)x

is

h

k 6 C(K~@){ P} c

_c c ( K ~ @ ) C ( A ) { q } = C(A)C(K~@){q}._ Let

There

to show that

k ~ c(~uA)x by

(V~ -~c

since

k 6 C(A){h}.

~UA ~c

We have

seen

(q,r,K)-small.

4.9.2.)

Now we t u r n to the proof GCA

and a s s u m e

that

LC~

be such that

S(L)

d S d {xEA

every

of 4.9. element

IL~HI<~.

Let of

~ E G w s n~ G

be g e n e r a t e d

is i-small.

Let

by

HC~._

Let

Define

: (VqEI ~ ) (VF C

~) x is

(q,F,~L)-small}.

--W

We

shall

prove

the f o l l o w i n g

(i)

S

(2)

C_~Dm H _c S.

(3)

(VzeIg

is c l o s e d

Proof Then

of

under

S)(VqEI ~ (i):

statements

x,yES.

c(~)y

by

@ c( uA)X Proof

of

Also

~uA --~c ~ @ .

(2) :

Let

therefore

IAxiaL I>_w

for e v e r y

qe! ~

and

k @ c(f~)x

x E G ~ D m H.

Then

IIz~III<~. ~,

F c

k @ c(~)y,

Let

shows

F C

and

--~c ~ A )

This

by

{q} ~ z. -

qEl g[

k E c ( ( 8 U A ) ~ L ) { q }.

~ --wE g,~A. by

(e)

~

be arbitrary.

~ --~c (iN@) h ~ c(~)x,

(3A --~c ~ ( F U O ) ) ( 3 k E C ( A ~ L ) { h ) ) ( ~ q and

c

Let

(30 --we ~ N F ) ( 3 h E c ( O N L ) { q ] ) ( V

8uA --~c ~ F ,

S.:

+.

)(38 c a~L) --~

Let

about

since that

xES,

x

is

Then

Then

k @ c(~)(x+y), by

and

yES.

h e C(A){k ~

IAx~HI_>~

x

by

~ --we ~ ( @ U A ) .

Therefore

since

bv~

x@gm H

k

and thus

h x+y~S.

and

(q, F, Ax~L) -small

is

(~'~)-small by

x

is

xEG.

Then

--0a

x~S

since

every

x

is

(q,F,K)-small

then

(q,F,M)-small

for

M~K.

Proof and

if

yES,

of

(3): by

Let

(2).

Ec(0--L){q}) (V~ --wc aN0)

zelg Let h

S.

@i ~

Then .

@ c(~)y,

Then by

z~c(F)y (~@ C y~S.

for some

--~c

aNr)(3hE Then

h @ c(F)y

since

4. I0.

~99

F --~c ~ 8 .

Then

N o w we Q:I ~

.

qEl ~

show

Only and

h@z, that

the

condition ~ c

~.

(3@ --~c ~ ( H U O ) ) the

hCc(@~L){q}_ conditions b.

of

4.9.)

Remark

Let

V

(i)

x

is small

(ii)

x

is w e a k l y

(iii)

x

is i r r e v e r s i b l y

(ii),

let

~,

let

and

let

and

4.10.1.

G

~JL iff

in

small

x =

with

q E C(F)OG. UGeB, such

proved

IKi~.

~

(VF c

(i)

zCIg ( ~ ) (G~DmH),

and

holds,

with

(2)

and

so 4.7.2

then yields

and

let

xCV.

(iii),

(iii)

Consider

V.

(i) ~

Thus

not

~)c

-~

OG

(Vk
xnc(

If

q @

Let

of

x

some

of

In fact,

(Vk>n)qk=O]},

is small

and

of

xEG. are

GCA.

Sb V

Indeed, Sb V.

but

not

Let

let

G

G

be

be a set d OG E B =

qEV,

and

of

(VxE

F --~c ~, Assume

L ~ KAAx.

@ --~c L

UG

Let

with

are done.

By h e r e d i t a r y that

elements

Assume

let

t h e n we

there

proving of

y:O]

V.

C(F)UG

x

(i).

disjoint

is w s m a l l ,

y E G~{x}.

version wsmall

in

UG

for

r)

elements

elements

that

p ~ C(F)UG

every

(~)

small

wsmall

@

small.

is w s m a l l

By w s m a l l n e s s

not

in

but

but

q e C(F)X

a strenghtened

However,

-unit

: (gn~)[qn~O,

To p r o v e

p ~ C(F)X.

p ~ C(Fuo)y.

(ii),

disjoint

Then

Then

ILI~w. that

=

disjoint

K _c A(UG)

b.

H~,

to be a set of h e r e d i t a r i l y

a set of h e r e d i t a r i l y

: xEG}.

Thus

: (3nEw) ( V k > n ) q k : n } .

(VX,yEG)Ex#y

n{DmAx

by

satisfied

V.

small

{qE~w

is said

of h e r e d i t a r i l y

(3).

Crs

in

is i - s m a l l

eG) J A ( U G ) ~ A X I < ~ .

Let

(3).

V.

(iii)

y

immediate.

are

of

below.

y : {qEe~

i-small,

by

be a

(i)-(iii)

(i) ~

4.7.2

zEIg(~)S(HOO)

_~ z

conditions

Then

of L e m m a

the p r o o f

4.9.

QED(Proposition

4.10.

is not

Then

c(@){q}

conclusion

complete

Since pE c(@){q}

disjointness

is w s m a l l .

We h a v e

4.10.1.

element

c a n be o b t a i n e d

from

small

ones

200

4. I I.

by u s i n g

4.10.1.

x = {q~2 then

and

z

Propositio~ and 4.9.

About (15)

of this

: (3n~)[qh:qn+l:l

y

4.7,

An example

4.11,

For more

4.11-4.13

see s e c t i o n s

below

(V~)

construction

see

there.)

(V~2)

is g i v e n

Prop.4.11

by r e l a t i v i z a t i o n

element:

if

O~{y,z]CSb_ x

disjoint.

of P r o p o s i t i o n s 5 and

given

(~ ~

In [ H M T I 3 5 . 6

in [ H M T I ] 5 . 6 ( 4 )

cs~eg) (~ ~ H 6 % )

in P r o p . 4 . 1 l

below.

a zero-dimensional

can

[Zd~l>2.

(Actually,

a l s o says that r e g u l a r i t y

with

4.6,

6.

[HMTI75.6(15).

that the c o n s t r u c t i o n

to s h o w t h a t

~cs~egmDc destroyed

applications

wsmall

Indeed,

4.13 b e l o w are a p p l i c a t i o n s

it is a n n o u n c e d

This m o d i f i e d

following

, (~k>n+l)qk:O]}.

are not h e r e d i t a r i l y

Propositions

be m o d i f i e d

is the

can be

element;

see

Prop.2.2(iii).

Proposition ~& ~W

4.11.

cs~eg~mc ~

~

Let

le--Hl~w.

~kw

and a

( G w s rC ~ e

Proof.

Let

g

~Zd

,

~

saw

and

h

: w >~ H

Let

and [~

IZd~w4)[l>2.

~2.

Let

HC~

be o n e - o n e

and

~a2.

Let

Ri

{q6a~

: HI qEHx (pi)

~ ~ [~

Claim

4.1~.i.

Proof.

Ax : H,

Then

[Hl:w

be such that

and onto.

Such

and

H,h

Let

and

~{ and

( H ~ h ~ i ) l q C_p i } .

[ ~ ( & ) {x}. R

w

4AEcsregnDc

= {q6~

p

pEW+l(H~) e x i s t by

and be IHI=w

and t h e r e f o r e

QEZd~

.

Let

Note

x d= U ( R i

: iS~}.

that

(Vi<w)R.eSS 1 : HI qEH~(Pw) }.

.

is r e g u l a r by u s i n g P r o p . 4 . 7 . E H ~ ( p i ) }.

are an

Define

an d

U{R. : i<w} q Sm s 1

There

such that

and

( V i < k ~ w ) p i ~ H (pk)

Let

1~

be a r b i t r a r y .

a

such that

iSw.

~A2

~EDc Let

G ~ {x}.

by

l~--HIkw.

Q ~ {qE~ Then

~Gws

We s h o w that

: (3i<~)H1qE nOrm

is

.1 2.

201

generated We

by

show

that

~.

If

F c

G.

Condition

condition fCQ

(i)

(ii)

t h e n we

of

4.7

is s a t i s f i e d ,

is s a t i s f i e d are done.

by

x

Assume

and

f~Q.

since

Q.:

IGI=I.

Let

Then

f~R

--W

g

we h a v e

Hn~ _c h*n.

Then

Pn[F/g]ex 4.7

iff

iff

FIgCF~.

Let

n~

F~AR n = O

and

therefore

for

every

FIg~F~.

F --we e,

q ~ Q ~ c (F) x. IKI~

by

By

pneQ

the

show

that

x

q~Q.

~2.

for

Let

so t h a t

x

to the

proof

Gws

-unit

rlwEHO ~

and

therefore

= xAW = x~

: R

= R

is not

4.12.

condition

Let

ke~{Pn(i)

We h a v e

seen

(since

IKI~

are done. L~O

}. x

Suppose

by

Such

that

Ax=H).

KCH,

(ii)

a

k

is Q - w s m a l l .

Then

~J~

is

In

there

zero

dimensional

modified

is a

~w

and

and

IeNHIAw.

ARw:O,

regular

~

and

have

W

Therefore

that

~W ~

is a

WE Z d ~

... comn : rlw~'~eGws~ ~nH~

showing

we

Then

therefore

[HMTI~6.16(2)

it is s h o w n

Ws

a homomorphic

having

for

u~2. Let

be the

Let h

Cs

since

The

.

.

Thus rlw(X)

rlw(x)

IZd(~ W~)

:

E

I>2.

Since

~ G w s [ eg.

and H

base and

image given

let

Hc~

be o n e - o n e

~

for

construction

qh(n+l)#Ph(n+l),

with Ax:H

. This

p~,

: ~ >--

that

construction

csregnDc

: (gnEw)Eqhn#Phn,

~J[ECs n D c

~,

W ~ ~•

: HI qEHx (pw) } ~ W.

elements.

to w o r k

Let

base

let

4.11.)

~2

~

that

t h e n we

that

we h a v e

L { K~(~Uh~n).

Let

4.11,

~W

since

QED(Proposition

{qEe~

i~L.

of

with

: {qE~

Zd(%~ W ~ ) ~ { O , W }

Remark

Let

such

g

is Q - w s m a l l . :

is r e g u l a r

compressed

rlw(x)

shows

q ~ c(F)x

i qk @ c(F)x"

Now

above

be

4.11.1.)

Returning

let

and

by P r o p . 4 . 7 .

QED ( C l a i m

E

If

(9 ne~) q~c (F) R n .

IFUh~nI<~.

is d e f i n e d

regular,

We

and

Then

and

exists x

fCF/g3~x

is s a t i s f i e d .

let

and

and

W

every

of

fEx

and

IeNHI~.

be

any

~

with

more

there

than

two

can be

is as f o l l o w s . such

and onto.

that

by

IH[=~ d

Let

(HNh~(n+l))lq generated

and

x =

~ p]}, {x}.

~J[ is r e g u l a r

and

Then

by P r o p . 4 . 6

202

4.1 3.

since

x

is w s m a l l

nearly

the

same

Proposition

Proposition such

~/f,gEp)[f#g

K

a

and

).

~2.

I~I

:

Let

P

such

that

IKI=2 Y,

O~K

Let

Such

a

there

it.

is an

the

B:Sg

Zd J~ ,

p cH~

be

largest

~

hence

such Let

~eLf

IH[=I~I that SCSbP

(VseS) IslAIHI.

that

csreg~Dc

possible

that

be

exists.

such

~ : H >~

(Sb H)~{0}

and and

be a Let

or

.

IPI:y

KCSbP

s~x=0].

(VsCS)v~:(s)=Sb

w

Let

is

of P r o p . 4 . 9 .

(VxEK)(Vs(S)[sCx

: P ~ Sb H

IZd~l>2

omit

i.e.

and

and

we

be such

P

LSI:y

(3~H~)

Then

HC~

IH•

(g)].

~

~A2.

Moreover,

and

Let

of

therefore

= I (a~)21,

H

exists.

proof

is an a p p l i c a t i o n

~ f q

that

The

6.~6(2)

~m

Cs

y ~

of

partition

a

below

Let

~w

Let

IaNHI~.

[HMTI]

IZd~l

from

Let

such

4.13

4.13.

(obtainable

be

as in

(~H~)

that

Proof.

and regular.

Such H.

w

be o n e - o n e

and onto.

Define

N

: P -- Sb H w

= < {jEH N d

as

{qE ~

: (3fEx) (H~m(f))IfCq}._

where

4.13.1.

Proof.

For

N(f).

zEG

Therefore

xeK

: xeK},

~vx d=

let

and

~{

[~ ( s

Yx

F --wE ~,

and

qi ~ c ( ~ ) Y x

hence

the g e n e r a t o r

QED(Claim

~[

Az:H

by the

LCH_

be

fEx.

for e v e r y G

of

is r e g u l a r ,

s

since

I~--HI>_~. conditions

for e v e r y

for s o m e

Then

Therefore

f][EDc

is i - s m a l l ,

.

we have

N o w we c h e c k

(HNN(f)) ~fC_q

Let

every

G { {Yx

6[ E c s r e g n D c every

is r e g u l a r .

q 6 y x,

For

~ : [~ ~ .

Claim

that

: ~(j)c~(f)_ } : fEp >.

xEK.

Clearly,

every

of P r o p . 4 . 9 .

Let

xEK.

such

that

Let

ieLN(N(f)uF)

ILl_>w.

~ --wc ~--{i}. satisfies

(~xeK) (Vi~H) (]fEx)i

Thus the

: ie~,

qEy x

and

let

Yx

Let

~

d=

G

to s h o w Let

we h a v e rE~{f(i)

is i - s m a l l

conditions

~/I.

of

A(Yx)=H.

By

by P r o p . 4 . 9 .

zeG}.

We h a v e

Then

4.13.1.)

I d Ig{ (ciz)-z

element

@

}.

and

of P r o p . 4 . 9 .

4.1 3.

203

Claim

4.13.2.

Proof.

I Z d ~ l = 2 Y.

Let

F c

H.

Let

HF ~

{jdH

: u(j)

D OueF}NF.

Thus

HF#O.

--W

Let

R F ~ {q6~ Let

xEK

uF))If~q

and

for

r ~ N(f).

let

some

Let

EH) Ep(j ) _D U ~ H F I f C q.

: (gfdP)HF1f~q}.

q~C(F)YxNYx . fEx.

By

iEF~N(f). F ~ ~(j)

I.e.

First

qER F

By

qqYx ~(i)

_~ v(f)~.

I.e.

fep.

show

that

qec(F)y x we

Then

by

we

have

~ v(f)

( V z d G ) c ( F ) z N z ~ R F.

we have

(H--N(f))If~q. and

seen

Therefore

therefore

H F _c H ~ ( N ( f ) O F )

We h a v e

(HN(N(f)U

(Vj e

and

(Vz~G) (VF c

therefore H)c(

)z~z

c

Rr9 This

implies

z~c(~)~ {zj

{ c i j z j N z j : j~J},

: jEj}

z~c(F)~

~ G.

Let

seen

x,wEK

by the

F --~C H. some

Let

sES.

q

and

Then Yx/I = 2 Y. then

by

v(f)

About

Notation: the c l a s s

We

show

for

q~

be

that

•

Now

Let

(yx@Yw)

G'

{ of

I

Now

show that

Then

{ij

: jeO}

_c ~,

(VjEJ)c(F)Z.~Z3 J _C RF

some

fCs

{iE~ by

I~,

that

by

To this

H ) Y x N y w ~ R F.

we h a v e

: Sb H.

Then

: q(i)=f(i) }=~N(f).

yx~y w q and

and

Let

sex--w_ for

~(s)

fdx~w,

Such

q @ RF

by

jEH F,

I.

therefore

: z~G).

Then

we

G'CZd~ " .

have

y x ~ y w q I.

(VF ~

x~w#O

seen

q

{z/I

zeI.

Then

By

qEyx~y w We h a v e

By the d e f i n i t i o n

QED(Claim

to

D ~(j)

(Vx,w~K) Ex#w ~

IZd~51

x--w#O.

such

IJl<~,

Let

z ~ c ( F ) R F = R F-

be arbitrarw.~

q(j)~f(j).

# Yw/I3.

~ --~c e,

(Vz~G)Az:H.

therefore

suppose

as follows.

: jcO}U~).

by

it is e n o u g h

jEH F

Let

exists

IHF]~w

above

for some

: jEO],

and

z 9 RF,

F ~ Hn({ij

above,

Then

j 6 N ( f ) n H F. a

Let

{C(F)Zj~Z j

as we h a v e

end,

(VzEI) (3F -c- W H)

(Vx,w~K)[x#w

G'~B

and By

IG'I

IBl ~

:

IAl

~ IGI : ~ 2Y

= 2 ~.

4.13.2

and

Proposition

Theorems4.14-4.15

Let

K

of u n i o n s

be

4.13.)

below

a class

of d i r e c t e d

see

of

EHMTI34.8.

:similar a l g e b r a s .

(under

c)

nonempty

Then subsets

Ud K of

denotes K.

204

4 . 14 .

In [ H M T I ] 7 . 1 1 Theorem be

4.14

replaced

Theorem

it is p r o v e d

below by

4.14.

shows

that

Ud (I c s r e g n L f

"reg"

cannot

that

"Dc"

in this

Let

~.

Then

(i)-(ii)

Ud(IcsregnDc

)

~

ICs

.

(ii)

Ud (I C s

nLf

)

~

I Cs

.

4.15.

Let

(i)

Ud (I Cs nLf

~w.

be d r o p p e d

and

"Lf"

cannot

theorem.

(i)

Theorem

) = I csreg~Lf~ ~"

Then

below

(i)-(iii)

hold.

below

hold.

) = { ~ELf

:

~(~)

is n o n d i s c r e t e

and

simple

or

) =

:

~(~j~)

is n o n d i s c r e t e

and

simple

or

is n o n d i s c r e t e

and

simple

or

IAIS2}. (ii)

Ud (I Cs ~Dc IAl~2}

(iii)

{ ~)LCDc

iff

Ud(ICs

)

~2 w .

c { ~ E I Gs

: ~(~)

IA]~2}. To p r o v e which

are

question show in

Theorems

closely under

that

has

(i)

Let

~ ELf

(ii)

Let

~@Dc

(iii)

has

~2 ~

Let

HC~

ElCs (iv)

For

EHMTI]7.28-29 a

Gws

hypotheses

the

following

and w h i c h

deal

is i s o m o r p h i c on the

~w.

Let

~CGws

a characteristic.

.

need

Gws

propositions with

to a

the

the

Cs

.

criteria

They

given

Then be

~][EICs

finitely

be

Then if

(i)-(iv)

~JL

generated.

non-discrete below

S2 I~1

has Then

and

hold.

subunits.

~ ElCs

if

subunits. be if

every

such ~

~

EGs--I Cs~ a single

shall

improved.

Let

~

we

conditions

c a n be

4.16.

that

to

additional

EHMTI]7.28-29

suppose

related

which

under

Proposition

4.14-4.15

and such

element,

that

has Hc~ that

I~Hl~w

~2 l ~ H l such ~(~)

and

(VxeA) IAx~Hl<w.

Then

subunits. that

~>2 I~NHI

is simple,

(~xEA) ]Ax~Hi<w

and

there ~

~

is an

is g e n e r a t e d

has

~ & by

(2I~H]) + subbases.

4.1 7 .

205

Corollary

4.17.

(i)

Every

nondiscrete

IAi~2 (ii)

nondiscrete

racteristic Let

be

Then

Problem

~

4.18.

of

4.16.:

4.16(iii).

((i)

is f i n i t e l y with

some

Proof that

of

4jr the

set

{~y(pi). l

and

(Vi,jEI)

that

IeNHI_>W

show

D[el Cs

by

and 9

: an(qi).

Such

CHMTI]6.2.

Let

eI)H1riC_qi, see that hence with

n>1 unit

3.15.b) every

J

{r.l

and

j6J~I

a homomorphic

: ieJ}

be

]yij

let image

Let I~Hi

to

(ii)

H=O

with a a

Cs

follow

and

if

conditions

~][EDc

of

(iii)

a characteristic.

of

assume ~

for

that

is such

ieI.

Let

~][(~Yi(pi))eA

by

LYiJ=n.

{r i

: i~J)

c an _

~

Then be

Gws

that

I#O

Hc~

be

~i

la~Hi_>~. to

= ~n(qi)

with ~i'

unit ieI.

For /[i"

every Such

and an (rj) Such

~i

j c PieI Iaj,

let

~j

exists

(Vie To and

by Cor. For ~3.

that

exists

/[i'

~i~Ws

(Vx~Ci) iAx~HJ<w. be such

to

where

that

ieI

9

such

We have

~

such

Assume ~@

1II_<2 I~NHI

exists

i ~i

some

?

from

~ r .q~n (ri) ~ and U { ~ n (ri) : ieJ} = an. 3e x i s t s o b s e r v e that /3[ is n o n d i s c r e t e

~jeWs of

and

the

Suppose let

isomorphic

H1riCqi,

.

iAi~2

isomorphic

we m a y

subunits

n d

III<2 I ~ H I ,

~n (ri) since

of

icI

and

an~

Gs ADc~

where

have

[HMTI37.21

Let

~i

(Vi,jeJ) [i~j

such

By

every

(iii)

satisfies

(VxeA) IAx~Hl<w. For

Cs

.

4.16(i)

{AeGws

: ieI}

iYil=rYjl .

to a

a cha-

EHMT]I.II.4.

Let

is n o n d i s c r e t e .

of

~

with

and be n o n d i s c r e t e .

many subbases

case

by

4.16(iii):

~eDc

nondiscrete

Statements

then

J~HI>_~

Cs

and w i t h

~w.

is i s o m o r p h i c

to a

finitely

is a s p e c i a l

Has,

if

(Vx~A) iAx~HI<~

Is e v e r y

~_>w.

generated

Then

i ~i

Let

,

generated

[~H[~w,

~_>~.

with

Cs

IAIS2 W

is i s o m o r p h i c

and

a characteristic

a characteristic

that

Let

characteristic

Proof

have

such

to a

finitely

and w i t h

U[~CA

Hc~

with

is i s o m o r p h i c

Every

(iii)

~ELf

by

is

EHMTI]

206

4.1 6.

7.27,

n<w,

I#O.

I _c PjEj have

Pj~j

! PiEI

By the choice of

and by the choice of

~j ~

I c

.

~nCCs

I ! PjEj

IHI:I~I

Let

~]5 be the

Then

L%elGs

Gs~

2 I~HI

be such that

1

has ICs

we

~

I c

GCH,

G#H

such that and

x ~ {q : (3i~G) CqE~Yi u{~Y

: i~G}

1

and

and

~(~)

For every

by

i~G

H1q~pi]}. by

x.

Ax:H.

We

d Yi :

let

: (Vj~H)qj

=
elements,

such that

nonzero

Cs

and

is simple,

(Vy~A) IAy~HI<~

In every

Pi ~

and generated

Yi = { q ~ Y i

Thus

I~I>2 I ~ H I

with finite base there are

IGI>2 I ~ H I

implies

~@ICs

.

4.16.

It is enough to prove

since

(Vx~A) IAxNHI<w

Then

Dc

c

~EDc

IGws

or by CAN1?.

,

# O

Let

A[VIw:o

and

~IAt~2 la~Hl

by

Let

2~

la~Hl~,

~[~

of l

i : B ~ I

(pi(a))

[AGN2]

Let

then

with

where (VaEB,~{O})am

Then

rl~Is~ Now

(VxeC) IAx~Hl<w.

~EGws a

Then

too.

4.17.)

Proof of 4.15.:

Let

o,_>~. The inclusions

c

~

"

~ ~ ~3[W~.

~elCs

Hc~

~ ~ ~

together

: i~I}

a~B} :

I~NHI~,

and

Assume that

be such that

i(a)

~ O.

~ECA

and therefore

~ V ~ u{aY! pi)

(~aEB~{O})aAW

By

Let

IAI~2 la~Hl

Cor.3.14(a)

=

and

(iii). and

W d u{~y

subunits,

by 4.16(iii).

by

Let

: i~I} : S u b u ( ~ ) .

QED(Corollary

by

Yi ~ {,< l,i>}

k~H~G.

Yi~A

9

since

: jEj]

be such that

subbases,

Let

: k,j~H}.

I~NHI~,

naYi(a) (pi(a))

Hc~

is a set of disjoint

is nondiscrete.

{~y!pi)

.

Then

Proof of 4.17.:

[HMTI?7.16,

let

Let

such elements.

QE__~D(Proposition

E Gws

i~G

(21a~HI) +

: i~G}CA_

@i~ICs

there exists a

with unit

~ q l Cs

(Vi~G)y i ~ ~{dkj only

and

by a single element

ci(x-dik)-dik. {Yi

~

(i/< l,i>).

with

have to show

Then

Let

For every

: ~<~)

Therefore

{rj

PiEI ~i ~

~j"

and therefore

I G I = ( 2 1 ~ H I ) +.

is generated

we have

By EHMTI~6.2

Proof of 4.16(iv):

<(O,i>

: jEj >

~j.

~i ~

Then

< ~j

in (i)-(iii)

hold

~ E

4.1 5.

207

because

~(~)

minimal

subalgebra

UdK=K

if

is s i m p l e

K~{Lf

of c a r d i n a l i t y

remains

, Dc

U d ( I C s nLf

generated

e~2 u

then

~@lCs

if

then

EHMTI]7.10.

~EDc

EHMTI]5.3, union;

and the

and further,

Every discrete

~Dc

#]~

simple.

subalgebras,

[BJ SIalU~=I~[ .

Now

Cs

is

~

is the

) = { ~Dc

~>2 u.

HC~

such t h a t

element,

is g e n e r a t e d : ~7~(fl)

~)

imply that IAI~2}

Then

and

f]~

~@UdlCs

element.

This

and

J~HI=~.

is s i m p l e

by 4.16(iv).

is n o n d i s c r e t e

~s

is n o n d i s c r e t e

be such that

~W(s

by a s i ngle

and if

and s i m p l e or

: T~

Let

Then

4.17(i)-(ii)

is n o n d i s c r e t e

~I Cs

) # { ~eDc

, ~(~)

generated

: ~(~)

by a s i n g l e

and

U~(tCs ~Dc

by

Let

Suppose

is an

is g e n e r a t e d

by

same in d i r e c t e d

Ud (l Cs n D c

IAI~2}.

there

~6Cs

N2.

) : {~ELf

s i m p l e or Then

}

u n i o n of its f i n i t e l y

is f i n i t e l y

and if

the

, lGs

P r o o f of 4 . 1 5 ( i ) - ( i i ) : directed

for e v e r y

since

shows

and s i m p l e or

IAI~2},

a>2 u . P r o o f of 4 . 1 5 ( i i i ) : Let

where {aj

Yo:{O,i}

and

: j<4}CA. Then

QED(Theorem

Let

has c h a r a c t e r i s t i c

Let

is simple.

2 by

Then

every

Then

of 4.14(i) :

HCa ieI

let

b i : 3 >~

of

Let

be such t h a t

I~22

~

Let

~c _

,

has characteristic 2, by E~4TI]5.4. containing

and let

subalgebra EHMTI]5.4

~EUd(ICs

s i n c e the c h a r a c t e r i s t i c

Let

~

: i<~>}.

~YoU~YI

{aj : j<4}

is in

tCs .

~.

finite dimensional

Proof

aj d: {<j

is sin~ple and

since no subalgebra of

P r oof of 4.14(ii) :

~(6~)

Let

w i t h unit

4.15.)

P r o o f of 4.14.:

locally

be the full G s

Yi:{2,3}.

~((1)

Therefore f)i~ U d l C s

~

#A aA~.

nLf

JHJ=I~HI=J~I.

I(~2).

and

EHMT32.4.64

)

by 4.15(i).

I

Then

2JLELf

, therefore But

be a set such that Let

x ~ {s~a3 and

Yi

Let

a n d onto.

and

~@;Cs

JAi~2II1>jIJ~221~I

Yi d= {,( i,i),( 2,i>} is o n e - o n e

be the g r e a t e s t

of

is 2 and Let

s

let

nil>2 l~i

: HIsEH2}.

b.• =d <( j,i>

V ~ u { ~ Y i : i~I}

For : j<3>. and

208

4.1 4.1.

let

~(J)

and

le~HIAto.

Claim

~ [~(~$V){Zix

4.14.1.

JCI.

Let

(i)

~(j)

E I Cs reg

(ii)

~(J)

E ICs

(iii)

Then

since

(iii)

if

~ (i):

JCGCI

u { H 2 (pi)

Let

then

: iEJ}

= 2 IHI = 2 I~l

Let

iEJ~

d= {sEC~ 3 d Yi = ~i x'

: HlsEH2(Pi)

zi

d U {Zi

: iEj} '

~([[a3){x

NOW every and

)eDc

below

}

by

(VieJ)A(~ix)=H

are e q u i v a l e n t .

iEJ.

union.

pEJ(H2)

Such a

p

IJI=2 I~I ,

be such that exists

by

IJl =

Define

~d ~ (~a3)(Z'l : iEJ}.

by T h m

1.3 s i n c e

iEJ.

r l Q E I S ~(J).

We show,

We c h e c k

z.

is a small

by 4.7(II),

the c o n d i t i o n s

regular

element

i,jEj,

rl z.eIs~T[, r l w E I S s i of 4.7(II) for each of the

A(B)z'I : A ( S ) w : A[V]o~ : O.

i~j.

that c o n d i t i o n iff

Then

x E S m ~t , {z i : i E O } c~ S_ m

pi[r/s]Ex~

and

show that condition characteristics We h a v e

c(r)zinc(F)z j = c(F)YinC(F)yj

(i) of 4.7 is s a t i s f i e d .

and

piEZi;

(ii) of 4.7 is s a t i s f i e d . =

IAYil

seen t h a t the c o n d i t i o n s

rl Z ~Is~T[, r ~ E I s ~ i Then fis

and

rlQEIS @[(J) by ],

:

and ~,

shows

U{y i : i E J } C Q ~,

~(J)

have

IH[ > to.

of 4.7(II)

are s a t i s f i e d .

4.7(II). Let

~(~fZi){zi})

: O

F C

(VfEx) (VF c ~) ( V s ) [ f [ F / s 3 E x --to

<){zI9 : i ~ J } c_ W

IAxl = IAzil

for

that

{Yi : iEj}CSm~)~(J) and therefore these generators are weakly small, too. Let and

H2 =

Q d u(ay.1 : iEj} r

Let

t h r e e cases.

Let

We m a y a s s u m e

) = xC~Zi,

and

~ E C s reg

IJI!2 I~I

~(G).

is a d i s j o i n t

: H 1 s E H 3 ( P i ) },

~d

~(J

(i)-(iii)

JCI,

~(j)c

Z. d {sE~3 l

W

Then

Ial N 2 I~I

P r o o f of

=

: ieJ}.

iEJ

such that

Therefore

and

f. d rl(Zi) o~i I. l fi(Yi):z i

ey since

~i E I s ( ~ '

~(~

i){Yi})

and

vi~ = b.x.•

Therefore

by

[HMTI~

5.1.

6.2

209

A[QZeY

'

= A[W3z

i

~Q

~(J)

= [~(~#Q){Yi

~Q

~ ( J)

~ ~W ~ .

~W

~

Proof

~

~

of

(ii) =

(iii):

~(J)

~W~=

%([[W){zi

~(j)

Let

~ ICs

E ICs rage

.

JCI

by

: iEJ}

i

: i6J}

and by

we h a v e that

~][(J) ~ ~ Q

be such that

The c h a r a c t e r i s t i c

is a set of d i s j o i n t

implies

racteristic

: iEJ},

Therefore

: iej} _c A(J)

IJI>2 I~I

Q = u { e Y i : ieJ} r W = u{Z

'

~(J)

e Cs reg.

h a v e to s h o w {Yi

= O

i

~(J)

q ICs

are t h e r e m o r e

of

nonzero

s i n c e in no

than

2 I~I

IJI>2 I~I ~(J)

is 3 and

elements.

Therefore

Cs

disjoint

We

with

nonzero

finite

cha-

elements

(by

~). Since

(i) ~ (ii)

QED(Claim

have ~(I)

we h a v e

seen

(i) ~

(ii) ~

(iii).

4.14.1.)

We r e t u r n directed

is o b v i o u s ,

to the p ~ o o f

set

~(j)

{ ~(j)

e

~ ICs a

: j -c- W I}

Ics~egnDc by

of 4.14.

,

~(I)

of a l g e b r a s .

by C l a i m

III>2 I~I

is the u n i o n of the

4.14.1

and by C l a i m

For e v e r y

and by

4.14.1.

j -c- W I

we

la~Hl~w. Therefore

But U d ( I C s reg~

~ m c a) ~ ICs a 9 QED(Theorem

4.14.)

5. H_omomorphisms

By [ H M T I ] 5 . 2 , we s h o w

the m e m b e r s

~[Ecsregows

However,

"nLf

(iii)

[HMT!35. 5 .

and

Proposition (i)

There

and

cannot

5.1.

Let

are a

of 6[

(csregows)~Lf

c s r e u~~ D c

and

Ws ~ S s

~ Dc

,

Ws

and a

~ Ss

by

"NDc~"

;

Below

([ELf

.

see C o r o l l a r y

~2.

simple. (ii)

simple.

s i m p l e do not imply that

be r e p l a c e d

ekw

are

.

Ws NDc

such that b o t h are

5.4

210

5.2.

Proof.

We have a direct c o n s t r u c t i o n for an

~Ecsreg~Dc

such that

is simple, but to save space instead of this c o n s t r u c t i o n we give here the following proof. ~ < O : i<~>

and

~(@@V){x}.

Clearly,

C l a i m i:

~

denotes the set of rational numbers.

V ~ ~Q(O).

x ~ {q6V

o ~ E W s ~mc

is simple.

cylindrifications. of

Q

: O:~{qi

: i<~}}.

~

.

The proof of Claim

i goes by e l i m i n a t i n g

C l a i m i here is an immediate c o n s e q u e n c e of C l a i m i

JAN73. T h e r e f o r e we omit the proof here.

By [HMTI37.13

~ E l C s reg

be a full

Ws

with base

directly indeeomposable QED(Proposition

Remark 5.2. Let

~w.

such that

p r o v i n g the rest of

~

is not simple.

and let

~(~ is subHence

~@Ss

.

(About r e p r e s e n t i n g h o m o m o r p h i s m s by r e l a t i v i z a t i o n s . ) There are

~ E C s rege

IZd ~ / I I > 2

and

and

IZd~f/Jl>2, Cs reg

and

,

for all

IEII~

and

Ws

is the following.

~EWs

and

and for all

rlvEHO~

and let

Then there are

q6x

(3F ~

~)f6c(F){q}.

since

rlveHO~.

{ rlv*~ . and

fE1 Z Nx.

Let

x:VNy,

A contradiction.

one could say that some

By

yEB.

A(~)x=O

~EWs Then

and

and

d {~0%6CA

V

such that

Let

~Ws

x q{O,1~

i~ ~ i~

, }.

we have

fEVAc~F)y = C(F)X = x,

A n a l o g o u s l y to e x t - i s o m o r p h i s m s ,

csreg-s are " e x t - h o m o m o r p h i c "

d e c o m p o s a b l e CA-s, w h i l e

Dind

Suppose

By 4.11

V6Zd ~I ~

The latter statement can be seen as follows. ~

JEIl fS

cf. 4.13 and [HMTI]6.16(2).

~EcsregADc

for some

I Z d ( r l v e ~ ) I<2

~EWs

rlveHO~ .

Notation:

~>2

5.1.)

IZd(rlv*4)[) I>2

However,

Let

Then by [HMTI]6.11,

and clearly

But a d i f f e r e n c e between we have

~.

(i)

to d i r e c t l y

Ws -s are not.

: IZd4il_<2}.

"Dind" is an a b b r e v i a t i o n

of "directly i n d e c o m p o s a b l e or o n e - e l e m e n t "

C o r o l l a r y 5.4 of 4.13 and T h e o r e m 5.3 b e l o w t o g e t h e r w i t h a part

211

5.3.

of T h e o r e m

5.3 i t s e l f

are q u o t e d

in

(6), (7),(~2), (15),

(~6) of

[HMTI]

5.6 and in [ H M T I ] 5 . 5 .

Theorem

5.3.

Let

~h2

and

(1)

There

is

~

cs~egnDc

(2)

There

is

~

Cs reg

(3)

There

is

~

Ws

(i)

H ~

~ I {~ECrs

(ii)

H ~

C Dind

(iii)

H ~

C I Cs

(iv)

H ~

C ICs reg

(v)

Cs eH ~

Proof.

Let

satisfies ICs n H ~

satisfies have

notations

Lemma

Sm ~

5.3.1.

and

Let

~

Dm~

Gwsn~

Let

@ol , ~ E C A

(iii)

Suppose

~=

~(~

~

H~

.

too,

(i), 1.2,

since

c ICs

satisfies (ii)

by (iv)

and

if

that ~<w

then

EHMTI37.21-22. too.

Therefore

(v) only.

Recall

If we the

1.3.1.

~

and

: l~Axl<~}.

) (am ~ --Dm~ )

and

4)[

(h(x)r Then

~

IAx~Ah(x) l<w)3.

Gwsn~

~

_c Gws reg.a

has a c h a r a c t e r i s t i c .

.

Proof.

Let

hS(Sm~

) c Sm~

follows

x E S m 4A

and

K ~ Ax~Ah(x).

(ii):

(iii)

and s u p p o s e

.

~ = 9 ( ~ )am~

c(@)h(x)

hold.

C Gws r e g .

Gws

)) (VxESm ~ ) E h ( x ) E S m ~

C Dind

~ECA

~E

then

from

~(~){xESm

P r o o f of

below

on s t a t e m e n t s

~:

and h e n c e

(i)-(v)

(v) t h e n

Suppose

Suppose

such that

HAw.

xk~

(ii)

(iv)

b e l o w hold.

satisfies

(VhEHom( ~ , ~

H~

(i)-(v)

and

(i)

Then

such that

HA~.

~Ae.

and

to c o n c e n t r a t e

b e l o w hold.

iff

and if

(iii)

(i)-(iii)

I b a s e ( ~ ) IS~}.

iff

Then

C I Cs

that

.

and

(i).

such

:

C Cs reg

•

~Aw.

and

Proof

of

(i):

immediately

= h(x) Suppose

~ El (Gws~ O m p ) r e g .

Suppose

Let

from

the

IKI2~.

Then

H{A_CDind~.

hEHom(~)[,~).

Then

definition

Sm.

Then

of

Let

(30 --~c K) e ( @ ) x = O

= O. 2Ji: ~ ( ~ ) { x E s m ~

: la~Axl<~}

and

~ E

2~2

5.3.

E Gwsn~

Then

~:

@~){xESm

~

: I~AxI<w}

by (i). Then

by Theorem 1.3 since every cofinite dimensional element is regular in Proof of

(iii):

characteristic. ~=

~(~)

Zd~

{)[ : [ ~ )

~EH~,

IBI>I.

Let

(Sm~ --Dm~ )

c Mn(/~) Proof of

Then

Suppose

and thus

(iv):

B = Sg Sm ~

nDm~ ) ~

~(~)(me

since every QED(Lemma

by

(i). Thus IZd~l~2

Then

: Mn(~)

since

~(~)

Let

~ = ~ ( ~ ) S m (g

by

(i). Thus by Lemma

6~ ~ D m ~ ) ~

~

has a

is simple and

by Lemma

1.3.3.

Then

is simple.

c I (GwsC~ reg.~ 1.3.3,

Let

~O

=

~ EH~.

~(~ ) (sm~

I (GwsC~ T h e r e f ~o. r e ~

(Gws~~

is regular

L5 .

and

~]~(~)

Dm~

n

,Zd~I~2

is simple.

5.3.1.)

We return to the proof of Theorem Proof of and

(Sm~ --Dm~ )

~

(i) and

(2):

Let

(Vn,mE~)[n#m ~ H nH :O3. n m

(VnE~)ITAH

I=i.

Then

5.3.

H e eSb~ Let

ITl=I~I .

be such that

M ~ URgH.

Let

d ~ y = I I

Let

(Vne~) {HnI~2

TCM

and

be such that

p E u

)

be

n

such that Sb R R

(Vi<j
d {qE~

Proof.

: T1q E u{T~ (pi)

Rgx C

Let

E (HnNK)NF

and

uE~, u#q(j). QED(Claim

: iEY}}

Sm ~ nDm~

Y~y,

is regular.

x : SbyN{O}

Let

Y#O. KCM_

Y~y,

and every element of

Y#O.

is regular

Rgx

Then it is easy to see that be infinite

j E HnN(FO{i})

Then

for all

q(i/u)

and let

for some

~ c(F)x Y.

Thus

A(Xy)=M

F --wE e.

ne~.

Let

Let q e Xy

in

and

i C and let

8 cic(F)Xy=O.

1)

LCSb ~

The existence ~(~){Xy

and

: (VnE~)lq~.~Hnl=l} '

Claim 1.

Let

R c ~

as follows:

Xy d {qER

Xy

Define

be such that of such an : YEL}.

Now

ILl>7 L

and

(VZ,YEL)[Z#Y ~

LZmYI
is a theorem of set theory.

2)i is regular and

MR

_c mind~

Let

~

by Thm 1.3,

~.

213

5.3.

Claim e=M

1,

IMfZ~

then

~

that

and L e m m a satisfies

(~IEII~)

{T (p) I ~ AnJ. XyEJ

iff

Then

Xy~I

iYm
~/I.

{Xy/I

Let

Crs

means

of

= ~. J~i.

(3):

Let

.

Since

]1~

We have

Let

~ {qEV

R

RCV

: I{iE~

and

Let

s ~

[~ V.

J ~ Ig ( s ){Xy

Claim

2:

Let Let

yCy

cicj{O } E J By

c(F)Xz].

Let

construction

of

iff

(VZ~y)O E Xz.

IZI
p,

there

(Vr~H)pj(m)#Pi(m)#O.

Then

Y,ZeL.

These ILJ>y

in

an a n t i c h a i n are m u t u a l l y

1~

facts

of disjoint

~ ~base(~)

satisfies

(i).

d

and

this

y = IVl 9

Then

y =

and

(Vj E

Such a

p

as follows:

YSY"

: YCy

i<j<~.

we have only Z5u

rYAZI
(ViEy)(3H~)[LHI~w

for all

(XyEJ

and let

since

XzUXw=Xzu W

Then

Let

and

: joy})}

Proof.

~

x : Sbu ~ SbR

: q!(SUU{pj

Y~y.

IYI 9

Y~y.

(Vi<j
Xy { {qeR

Let

Let

=

by

V d ~ (0)

and

: qi#O}INl}

Then

Since

be such that

W

IEIls

=

~ d ~xl,

Next we s h o w

and

contains

seen that

_

p E 7(~)

Define

i~I .

5.3.1.

of c a r d i n a l i t y

~

and if

: (3qey)T1q6w}e
of an a n t i c h a i n

I>u =

~eDc

Let

: (3qEXy)T1q6w}I

Xy. X z = X y N Z e I

Eye{i}) ( V m E H ) p j ( m ) # P i ( m ) # O ] exists.

: i{w6W

is an a n t i c h a i n

we have I>~.

1 and L e m m a

Ibase(~5)1>~.

and the m e m b e r s

Ibase(~)

Proof

and

~ ~ECrs

>y

then

and t h e r e f o r e i{wEW

: y6L}

~/I

cardinality in any

JYl=y

I~MIZw

J ~ {yGC

J6Ii~ since

by

show that

NI { ~ / I } )

Define

Clearly,

If

(v) by C l a i m

(V~ECrs

: pET~}.

5.3.1.

and

IYI
IYI
Then

and

xy~dij

Suppose

c {~

iYI:y.

to show

(VZSy)[IZI
and let

F c

~.

is an i n f i n i t e for every

HCa

mEH~F

(Vi<j<~)Xy~dijEJ.

Let

~i

: uE~l}

We s h o w

~

(VF ~ i6y~z.

such that we have

C

Xy@a.

~)Xy By the (Vj6Z)

O(m/Pi(m))

6

e Xy~C(r)Xz. QED(Claim Let Let

2)

LCSby

be such that

f][d ~ ( ~ ) { x u

: y6L}.

ILI>y Let

and

(VZ,YeL) EZ#Y ~

I d jnA.

IZnYl
By the second

= iZI3.

condition

214

5.4.

on

p

we have

{Xy/I

XzNXy _C Xzny"

: YeL}

is an a n t i c h a i n

(Vi,j<~) (VYEL) Ibase(~5) I ~

Xy/I

ILl>y

and

has c h a r a c t e r i s t i c

~

satisfies

done

then case

~

>

satisfying

of

such

for some

observe

that

~

. Then

satisfies

(i).

tha~ R g X C Sm ~ N D m ~

and then apply Lemma

(i)-(2)

of

5.3.1.

TC~

with

5.3, we o b t a i n

that if

(i)-(iii)

there

is

l~-~Tl=w w

N

of T h e o r e m ~E

as it I~I

5.3.

In

Ws NDc

5.3.

with Corollary

keeping

to a

satisfying

(i) of T h e o r e m

i n mind

5.4 b e l o w

see also

csregcDind

.

figures

[HMTI]5.7

In connection

with

(v)

4 in [HMTI~9.

5.4.

Let

ii)

H(cs~egNDca)

~2

cs~egnDc

H( Ws a

(v)

that ~/I

~]~ Ws

(v),

we do not know w h e t h e r

(3 ~ E

(iv)

~,

Ws nDc

i)

NDc

and

) IAI =

further

eke.

Then

) g Dind

~<~.

below

hold.

and

D icsregulws

is not simple

Then

(i)-(iii)

I (a~)21.

_~ Dind a _D I C s rega

(3 P.J~E~cs~egNDca) [ ~

Suppose

and

I~I

see P r o b l e m

iii)

(ii)

~

[HNTI36.10

Corollary

Therefore

is

In c o n n e c t i o n and

ILl in

~/I ~ ~ E C r s

the proof of 5.3(3)

in the proof

there

2 we have

5.3

By r e l a t i v i z l n g was

Let

by the above.

that

QED(Theorem

of c a r d i n a l i t y

~ dij/I.

To prove ~

Hence by C l a i m

(iv)-(v)

Dind N H ( c s ~ e g N D c a )

~ ICs a

Dind ~N Ws

~ ICs

.

and

H~

_c Dinda and

IA1>23.

b e l o w hold.

and .

x c s ~ e g ~ D c ~ _~ SWsa.

Proof.

(1) and the

second

part of

because there

~

(ii) Ws

first part of follows and

(ii)

are i m m e d i a t e

from i n s p e c t i n g

IEII~

were

by 4.13.

the proof

constructed

of

with

The

[HMTI36.16(2) 4A/I ~ D i n d

215

5.5.

and

it is e a s y

8.4

implies

the

in t h e p r o o f and

an

is not

that

result

of

such

~

Cs

simple.

that

C H Ws

and

consequence

then

(v)

if

a~w+~

for a n y

is n o t c o m p l e t e l y

if

if

[HMTI]6.i6(2)

IEII~

k n o w of since

to c h e c k

~EDc

(We n o t e

that

c a n be a m e n d e d IZd

4A/II

(iii)

condition

(i)

from

~

(i)

But

to o b t a i n

an

The

of T h m . 5 . 3

since

if

WE

EHMTI]

2~x<~

Ws NDe

amendment

implies

(iv)

then

the construction

is a c o n s e q u e n c e

IBIS2 I~1

then

.

= 2m a x ( e ' x )

obvious.)

follows ~EHxWs

e~.

then

of Thm.5.3(1) that

then

I Cs n H W s

is an i m m e d i a t e

of T h m . 5 . 3 .

Q__EED(Corollary 5.4.)

Later constrast IGws c ~

we

shall

with = I Cs

frequently

<~{TI37.13,

use

Proposition

7.17 w h i c h

state

5.6 b e l o w that

Ws

which

is in

c icsreg

and

.

iGws wd =

Gws n~

Cs

=

IGs

:

HSP

Gws

.

= IGws~ ~

CsnDind GwsCOmp

reg

csreg

Ws

Figure

Proposition Gwsn~176

IK

# IK reg.

5.6 b e l o w

implies

5.5

that

only

(~h~)

of t h e

seven

Ke{Cs,Gws c~

classes are

we

{Gws wd,

such that

c

216

5.6.

Proposition

%.6.

Let

~ ICs reg

I Gws c ~

regnLf

= IcsregnLf

inclusions Let

Proof.

Let

on F i g u r e

KE{Gws wd,

Gws n ~

~w

~2.

x { {q6V

6~EGws c~ e C s reg

and

since y#O

: H1q[0},

Let

~ E G w s c~

EWs~NLf~ Proof

Let

.

p61 ~

and

of

(ii) :

/AEGws

there

is

icsreg

.

Then

~

Now

Then

= I Kr e g

H,L c e

~ ~ ~x{l}, ~

= IGs

be such that

V ~ ~ x ( 0 ) u ~ x (~)

{ 9(@~V){x,y}. ~ q l C s reg.

and

a#O,

4 A 9 1 C s reg

with Thus

IZd ~ 1 % 2 ,

V ~ abase(~)

~ iGwsCOmp

b#O.

Then

Let

~E

Then

a.b~O

since

AxNAy:0,

~

is s i m p l e by

1%

= V.

Gws

thus (p)

Then

s

x~0,

~ ~/V6[

Thus

and

Ws

(i). Cs n D i n d ~

} ICs reg

_c icsrega

for some

V 4)[ ~

is a

~EPWs

e.g.

.

By

By

by 5.4(v).

IGws c~

: ICs

by

[HMTI37.13.

.

By [ H M T I ] 3 . 1 and

GwsWd-unite and by

I c ps

is proved.

= IGws

IWs

from

(~i<j
Then

c iGws wd

follows

by the d e f i n i t i o n s 9

I _c p ~

ili"

reg

in 5.7(iv).

such t h a t

IGws wd = IGws n ~

Proof

Let

and

AaNAb=O

[HMTI]7.17 ~

= Ui

Let

= IGws

(i) :

IK

We s h o w that

reg _c D i n d ~

~ cPWs~

~ I G w s wd. then

of

Then

: m]q!~}.

1.3

reg w i l l be p r o v e d

Let

~

and the i n d i c a t e d

by 3 9 15(a).

(i) in the p r o o f of

P~

y ~ {qEV

regnLf

W s uCs reg~ _c GwsC~

.

Gs}

0 ~ ~xl

be such t h a t

_c i c s r~e g N L f ~

iGwsCOmp

Ps

.

x. y=O.

[HMT]2.3.14.

by

Proof

by T h m

a,bEB

Gws,

(Vpea) (VqEb) p [ A b / q ] 6 a . b . and

= HWs NLf

5.5 are all d i f f e r e n t

HNL:O. Let

reg~Dc

and

but

hold.

IH]NIL]NI~--(HUL) 1 ~ , Let

~w.

regnDc

The c l a s s e s

(iii)

and

Gws c ~

(i)

(ii)

~2

~ ~ e G w s wd

7.16

implies

HS~ Gws

s

c Gws =

= IGs of

(iii) :

GwsWd

= GwsWd

reg

is easy to see by the d e f i n i t i o n s .

d

Now

[HMTI]7.14

proof.

~

[HMTI]6.2

Gws wd c Gws n ~

[HMTI]7.14,

P~

together with

(ii) of the p r e s e n t

theorem

complete

the

217

5.7.

QED(Proposition

Theorem between

5.6.)

5.7(i)

(i) and

below was quoted

(ii) of Thm 5.7 implies

making a homomorphic new zerodimensional we often

stated

motivation

Theorem

(iv)

elements.

Let

•

NLf

) ~ ICs

H( Cs

nLf

) ~ I {~ECrs

Dind NH( C S

nLf

) _~ ICs reg.

~2

ADc

and

Let

I E Ii~

= O/I ~ xi/I]

Proof of

and

in

(i): and

Let

IBl=p

Lemma

5.7.1 there are

~

The following

J ~ Ig (~ ) I .

ting (*)

it without

be complete x E PB

I c Ii~

BCA.

in

Some

and for all

lemma is well known

~.

from

proof. and atomic.

such that

and

Let

(Vi<j
~ ~ F~.

p=IBIAw.

Exi/I.xj/I

=

infinite

x ~ PB

such that

and ~/I.

such that

in

Let

4Ac~ then

Suppose

Ibase(~) I~.

(Vi<j
~/J.

d (~x p = I )21.

Let

and atomic

By EHMT]2.3.7

= O/J ~ xi/J]

~ECrs

HK ~ L.

: Ibase(~) I~}

reg

is a complete

= O/I # xi/I]

some

~<~

) ~ iGwsCOmp

~ ~ ~x

Exi/I-xj/I

Exi/J'xj/J

KCL

o~/I.

Then

Let

if

reg

eA~.

/~EBA

5.4 for two classes

) ~ iGwsCOmp

the theory of BA- s. We quote Lemma 5.7.1.

is to create

below.

H( Cs

Let

There are

into a non-ICs

~.

Cs nDind ~H( cs~egNDc

Proof.

Cs ~Lf

instead of the weaker

and

xCs nDind AM( Ws

The contrast

that the only possible way of

In C o r o l l a r y

Dind ~HK ~ L

(i)

(iii)

image of a

for this is Thm 5.7(ii)

5.7.

(ii)

in [HMTI]5.6(II).

Then

= O # h(xi/J). : I b a s e ( ~ ) I~•

BA.

(Vi<j
be arbitrary I : BNJ.

Thus

h E Is( ~/J, ~) Ii~

Then by

l~l~]
This proves

such that (Vi<j
~8

5.7.2.

Assume next E Cs

~<e

and

HeCs

w h i c h is impossible by

proved for

Let

(ii) :

e SPCs reg.

Assume

To prove

(iii)

Lemma 5.7.2.

and

Let

(i)

~

(ii)

Suppose

l~-Axl Then

@ I (Gws~~ ~

Let

hypotheses

of

and

if

and

~O

Suppose

h : ~

O ~ dol. Y-slY

and

h(y) c

~

~

{O,{qEl 9%

p ,q~Y thus

pO~qO,

The same a r g u m e n t works

and

{x,y,z} c B

and

for

Then

p~

if

~:0.

satisfies the

and

Then there is and

Y ~ h(y).

pO~qO .

h(y) E {O,{q61 ~ implv~

reg

: q(O)=u}}

p(O/qO) c y

AY _c i

~ EGws reg~

for every

~ ~EGwsCOmp

E G w s C O m p reg

(~u~base(~)) and

e I pCs

H.

H ~ Ax.

h : ~

is compressed,

Ax:H

Suppose that

x.z=O,

has c h a r a c t e r i s t i c ~

Let

Suppose

E y,

~

and suppose that

: (~iEH)qi=u}}.

Y-doi,

~ E I C s [ eg.

{x,y,z}.

(ii) below hold.

c_ {O,{q61 ~%

We have seen

0[E

Then by [HMT]2.4.43

Thus

(i) and

such that

p,p(O/qO)

IAI:I.

0 y.sly ~ dol ,

Lemma 5 . 7 . 2 .

Let

IZd~J%IS2.

: I~I,

uEbase(~)

Now

is

By [HMTI37.13-16,

be g e n e r a t e d by

~ IGws

~ECA

C l a i m 5.7.2.1.

~

~ / J q ICs

reg

E I ~Cs~

since

By [HMTI35.3 then

(iv) we shall need the following lemma.

~ECA

(ViCAx)x+z S Y-doi.

@Ae.

that is

i n d e c o m p o s a b l e or

i = Ay _c Ax : Az,

Proof.

(~) above. Thus

~Jt6H(Cs r]Lf ),

~ E D i n d @r

is subdirectly

Proof.

h e Is( ~/J, ~ ) .

H<w.

Proof of

and

and

Now

since ~ eGws

{h(x), h(z)} c

Then

AYc

p(O/qO) E ~

h(x) E {O,{qEl ~

1

is regular.

imply

: qO:u}}.

1

Then

Y's~Y ~ d01. (ViEH)x

: (ViEH)qi:u}}.

z.

QED (Claim 5.7.2.1.)

Now by C l a i m 5.7.2.1, i G w s C O m p reg

x-z=O,

x~O,

z~O

immediately yield

~

~

To prove 5.7.2(ii), we shall need the f o l l o w i n g lemma.

5.7.2.2.

First

219

to every

YS ~ { q ~ 8 ~6

ordinal

: qO=O}

~ [~(s

8k2

and

}"

we define

X 8 ~ {q~B

By this

the

is the class of all ordinals, 6cs[eg Lemma

for e v e r y

a

Cs reg~ ~ 8 .

: (Vi~H)qi=O}.

system

< ~8

Let

Let

~ =d

: B6Ord~2>,

has b e e n defined.

By Thm

8A2.

Then

~@ ~8

where

1.3,

and Ord

~

8A2.

5.7.2.2.

(ii)

~6

~ I 8Cs~

Proof.

Proof

ordinal

p

: ~xPS/F

~ PB

6he.

for e v e r y

BeOrd~2.

(i) :

Let

be an

UdcFEIS(~6,Z~)

Let

= IP~/FI

where

By

-choice

F

exists.

function

: j

.

~ EbC s

~+Ecsreg8 ~

be an u l t r a f i l t e r

Such an

[ =
in 3.12.

for some

F

.

(F,( B : i

,e)

c(i,[/F)=~

was d e f i n e d

6Aw.

IPB/FI

such that

(rueS) (ViEe) E H o ( ~ 6)

of

for e v e r y

reg

Let

c :

such that

The h o m o m o r p h i s m and

~

on some

such that

EH M T I ] 7 . 6

base([:)

UdcF E

7 ~ 12

'

!

: PB/F.

Let

(VqE~U) ( V i ~ H ) [ q i = O / F

iff

U = PSlF. U d c F ( Y 8) = {qE~U

: qO=O/F}

U d c F ( X 8) = (qE~U

: (VieH)qi=O/F} ,

c(i,qi)=~].

Let

y d=

and

IUI.

because

Since

R +B = Sg{UdcF(YB)

have

~+ ~ ~ Thus ~8 ~ ~ Repeating B y" 7" we o b t a i n ~ ~ ~ . Hence ~B ~ ~ " w

Proof

of

Let

Pi q Tx(pJ)

i E ~i

let

iE~.

T ~ ~H. Such

V i { (qE~

Then

a

rlviEHo(~x, TAAv

vAV'#O'I

Thus

rl(Vi)EIs( ~

and

by

we have

is e q u i v a l e n t QED(Lemma

is finite,

Pi~

to saying

p E ~(T )

exists

since

that

hence

and

~• 9 ~ ~

Let

argument

for

be such that ITI = I~I.

V O { ~•

for some

, ~i ) .

Vi = ~ ~

~

5.7.2.2.)

p

T~[i)

Then

i ( ~li) = V i

Let

: TIq~T~(pi)}.

E R ~{0}.

[HMTI]6.2

the above

we

7

(ii) :

(Vi<j<~)

Let

UdcF(Xs)}

Then

i : iE~1}.

~ [cE~G w s ~

by r e g u l a r i t y ~

~

. of

I _c PiEe

{V i : iE~} -c Zd~@~

I -c [[e~. ~ I H Cs

.

Hence

~ ~

For all

Let v

vE

we have

]~i .

Since

are d i s j o i n t ,

I c ~@~

which

220

5.7.

N o w we t u r n to the p r o o f of 5 . 7 . 2 ( i i ) . has c h a r a c t e r i s t i c 5.7.2.

Lemma

I ~ Let

~

and

By E H M T I ] 7 . 1 4 - 1 6

P~%

for some

j~.

~IE

Then there

by C l a i m

{x,y,z}

5.7.2.1

Jcs reg.

exists

=

We have p r o v e d

Case

i

have

Assume Bj:~

~#0.

for all

u~U.

By

~lj).

the e x i s t e n c e

we may

~

~

of

~ ~

assume

J=~.

U d base(~j).

: q(O):u}, I _c

~B

of

B~Ord

Hom(~5, ~(Bj))#O

Hence

,

and h e n c e

Let

~lj ~

~EtGws

the h y p o t h e s e s

reg

IBI_<~

Hence

T h e n by

j~.

satisfies

~j c_ 9 { { q 6 e U

for some

that

~6SPCs

baHo(~,

we h a v e t h a t

I9

c B

we h a v e

: (Vi~H)qi=u} } Ibase(~j)

Suppose

{q~U

for

Then :

~ : such that

and by E H M T I 3 5 . 3

I _c ~ ~

e l Cs

,

we

by L e m m a

5.7.2.2(ii) . Case

2

Assume

Bj_>~

~=O.

for all

je~.

Thus

QED(Lemma

~

=

je~. ~

By L e m m a

I c ~ ~

(iii)

I~I 9

p

and

(iv):

Let

LCH

~ : L1qCO_

d ~ z = {qE

p

and

: (H~L)IqC_O

and

and

#]LEcsreg~DC {x,y,z]

H1q_CO}.

nI=O

since

{x/I,

y/I,

Let by

s

Let

we h a v e

~( B j )

for all

5.7.2.2(ii).

~EH~)~.

Let

(z~V)/J}

ILINIH~LI>-w.

0EH,

IHl_>w

and

We let

: qi#O}l=l}.

d ~@~x. by T h m

{x-dij , z-dij , x-y,

satisfy

and

~)w C { q E ~

reg

~V ~

where z-y

: i,j~H}

of L e m m a

So far, Then

j = B N I g ( ~ @ V ) { v N v }. satisfy

1.3 and L e m m a

: ( H~F)IqCO}.

the h y p o t h e s e s

~d

and

I d ANig(s

E I Cs ~; (GwsC~

V d ~ (O)

(ynV)/J,

~p ~

be such that

: qi#Oi=l}

I {i6H ~

(VwEI) (3 F c z/I}

H c ~

H/A c D i n d

c Sm 0~ .

Clearly

Let

i{iEH

Now

yields

argument

~ d ~•

where

:

then

by L e m m a

o~

be s u c h that

~[ d [ ~ ( ~ ) { x , y , z }

since

T h e n by the above

5 9 7.2.2(i)

E t Cs

--

y d= {qE ~ : q ( O ) = 0 } , d x = {qE

p_>~.

5.7.2)

P r o o f of I~NHI

Let

~

EWs NDc

Again,

the h y p o t h e s e s

v = {q6~ c I

~/a

of 5.7.2.

and

Thus

5.7.2

(iii)

5.3.1(iv) : {x,z}n

f)L/I and

which

then

has b e e n proved. and and

H~

C Dind

{ (xnv) /O,

QED(Theorem

5.7.)

221

5.8.

Remark

5.8.

CHMTI35

imply (i)

~2

and

and the p r e s e n t

~m.

Statements

section.

Corollary

(i)-(v)

b e l o w h o l d by

5.4 and T h e o r e m

5.7 here

(I)-(2) below. Statements by

(2)

Let

Dc

(i)-(iii)

or

Cs reg

(iv) b e c o m e s true if

false

Cs reg

below become

is r e p l a c e d if

Lf

f a lse

by

by

H( c s r e g N L f

) c

I

Cs

(ii)

H( c s ~ e g q L f

) ~

I

Cs U o C S ~.

Cs AM( ~ c s r~e g n L f ~ ) -c

I

(v)

C s NH

Problem

5.9.

xCs~

~CS~UoCS ~

~Zw.

if

~

iff

~<m.

We k n o w t h a t the

5.10 are not e q u a l i t i e s ,

marks,

w h e r e we do n o t h a v e c o u n t e r e x a m p l e s .

H ~Cs

but

except

inclusions

Figure

6.9 and 6.10 look l i k e if

Dc

iGwsCOmp

= I Cs

those

reg

indicated

is i n c l u d e d ?

= I ~GwsWda reg

reg

/ H

?/ H W S c~

I

/

C s reg

1 C S ~ eg

/ 1 Ws a Figure

5.10.

(~)

indicated

GwsCOmp

reg

on

by q u e s t i o n

How do F i g u r e s

?=

H Gws cOmp

it r e m a i n s

.

_c i Cs U o C S

Let

by

.

D i n d AH( c s r~e g n L f ~ ) _c I Cs

(iv)

is r e p l a c e d

Cs.

(i)

(iii)

Lf

Cs.

is r e p l a c e d

is r e p l a c e d

if e i t h e r

[HMTI35.7,

222

&&.

5.

Problem in

5.11.

K

surjective?

(Vf,hEHom( We

Let

~,~

note

2S~. That

is,

))[B1fCh

that

this

Let

K E

are

, Gs

there

~ f:h3

problem

{CA

but

~

}.

c

Is e v e r y

~j~ E K

such

epimorphism that

(V~EK)

B#A?

is e q u i v a l e n t

to a p r o b l e m

in d e f i n a b i l i t y

theory.

6.

For lated

some

notions

sumption

(ii)

our are

that

purposes too is

H

be

a set

similar

to

Let

an ordinal

The

definitions restriction

instead

and

CA~-s.

let We

h

of

an

Let

CA R ~ { ~(n)0~

I }.

Let

H

and

: ~ECAIH LCT

be

an a l g e b r a

similar

same

(i)

as

in

of

CA

lies

and

d

in t h e

arbitrary

re-

as-

set.

H

be

from

4.7.1.i.

a set We

generality

an o r d i n a l . We

Let

n

note

that

n

: H >~

IHI .

sets,

h

We

define

an algebra

p'~

,-~

,O ~ ,1 ~

,

Then

: H ~ T

obvious

: H >--

since K E

and

let

~(h)~

~

be

to be

changes.

~L

the

~

IHI.

4.7.1.1

there

we

Recall

and did

{Ws,Cs,Gs,Gws,~ws

the

4.7.1.2

function apply that

rd (n)

to

the

not

assume

8

is

norm

_ comp _ wd _ creg ,~ws ,~ws ,~rs ,

define

d ~ A creg = mH CrSH

Related

be

=d < A , + ~

the

and

K H ~ {(rd(n))~(n)~ reg KH

~

~(h)~

CAT-S. with

and

.

Let

Crs}.

three

to

above

: H ~ ~

define

C h ( i ) 'd ~h ( i ) h ( j ) ) i , j E H "

present

(iv)

original

6.O.

{ ~(LIId)~ (iii)

the

restrictive.

~

Defin• (i)

of

Products

notions

: ~EKIHI}.

if

like

K H c_ G W s H.

BOH,

Nr H

See

etc.

1.6.1-1.6

are

9

defined

2.

analogously.

6 . "!.

223

Remark

Correctness

and

section

of

KH

n

: H >~

K

CA H

IHI

:

gives

Let

~ 6 K B}

of

all

UESubb(~).

that

if

a natural ,

then

rl(~U)*#J~,

UESubb(~L)

is v e r y

from being

property

different

[HMT]2.6.2 definition

enumeration

p : e >--

B

we h a v e

Reducts

: UESubb(~)>

one

the

be r e g u l a r .

the o t h e r

and

might

of

into

e Cs

be t e m p t e d

subdirect

below,

properties

-s

is a s u b d i r e c t

6.1 b e l o w

by the F a c t

Gs

decomposition

natural Thm

of

rl(~U)*~

subdirect

s o m e of

Thus

The

of the

and c l e a r l y

the n a t u r a l

true.

from

(iii).

decomposition

EHMTI]I.15

will

from

~,B

< rl(au)

or at least

in

choice

subdirect

of

follows

~ Hbase(~).

EHMTI]6.2

In v i e w

csreg-s

be as

of the

I~

then

~)i by

{)lEGs reg

far

K

by s e c t i o n

Crs H ~,

~EGs

decomposition

definition

for all ~ r d i n a l s

for any

EHMTI]6.2

yields

Let

since

that

-s.

Namely:

is i n d e p e n d e n t

= { r d ( ~ ) * ]~(~)~

Cs

above

Reducts.

and

We n o t e

of the

Gws

to t h i n k of

factors

states

that

regularity -s

for

this

is a

introduced

in

[HMTI]I. Fact:

Let

~eKE{Gs

rl(~u):~

eK

Proof:

Obvious

Theorem

for all

6.1.

(i)-(iii)

,Cs

,Gws

a~O3

and

x~2.

q

(ii)

rl(~U)*O[

~ Dind

(iii)

rl(W) * ~ ~ C s ~ eg ,

moreover

empty

that

I

d

Sb

O3

c~.

For

XH :d {q~e(~•

QED

There

is an

e G s ~ eg

for w h i c h

hold.

rl(aU)~:~

Let

Then

UeSubb(~).

(i)

Proof.

,Gws wd ,G w s ~ O r m , G w s ~ O m p } "

by the d e f i n i t i o n s .

Let

below

,Ws

W~2 ~

a->w any

ICs reg

such

and HEI

(< n,H>)

for

all

U~Subb(~

for

all

UESubb(~).

~>2. we

rl(W)::~

~ Cs n D i n d

for any non-

rlwEHO~.

For

any

define

: ne~

).

and

H•

set

s

let

s d <s

: i<~>.

224

6.1.1.

d

y = U{x H : Hel},

F i r s t we

Claim

V d U{~(Hx{H})

show that

~

6.1.1.

I~AzI<w

Let

zCJ--{O}.

Proof. be fixed. infinite exists

Since

a finite

that

AZD~.~L.

Claim

6.1.1.1.

Proof.

there of

ra m E Ism( ~ L Then

~,

z~B

d i P : g( n,K>"

Let

Y : (Hx{H})x{(@~L)lq},

Then

by 4.7.1.2(ii).

Let

=<(n,K),(~L)Ip>}.

~

be the

of

rbL(P)

QED(Claim

be fixed.

F

H~I

by

There

z E S g ( ~ L ~ J ~ ) {y}.

We show

rb p : V - R g r b p

be

and

Recall

and

T h e n by and Let

: kor b L ( q )

J[ .

with unit

We h a v e

Since

q e z N c ( F ) X H.

y+NLz

pEc(s

lZ

Then

: W >~

W

k(y+):y+ Then

qez

N1kCmd

w e have

rbL(P)

E rdL(Z )

y+NLY =

: (VjEH)hj: be such that and

Then

(W.~(YUZ)) IkC

~Is(~

Then

is

~(y+nLy):y+NLz

since

m:Sg (~[) {y+}.

rbL(q)ErdL(Z). Since

rb L

Then by

is o n e - t o -

implies

pEz.

Now

for some

nEH.

Let

K.

p E ~ ( H x { K } ) ((n,K>)

,~ )

because

6.1.i.i.)

p E z N C ( F ) X K.

pez.

Y,ZeSubb(K)

we h a v e

: {hELz

.

: j<~)

We show that

H=HAL:KGL

E ~ ( r d L ( Z ) ) : rdL(Z) "

by 4 . 7 . 1 . 2 ( i ) , by

Gs L

~% d

g _d ( ( q j ( O ) , K }

p e C ( r ) X K.

k

and

that

Let

I.e.

By 4 . 7 . 1 . 2

~ d ~(~me~){y}

(< n'H))

W:base(~).

rd p d rdL.

rd L : rbL~':.

Z : (xx{K})x{(~L)Ip}.

(W~(yuz)) IkCId.

i@L,

pezNc(r)x K

and

Let

gecic(N)X K

full

k(rdL(Z))=rdL(Z).

Let

fEV)

v + = rdLY. ~

Thus

V

for some

(VmE~) k (( m, H }, (~--L) ] q) : (< m , K >, ( ~ L ) I p)

C] (k~k)C_Id,

one on

:

q~(Hx{H})

Let

a base-automorphism and

H

rb p d rb L

: (VjEH)hj:((n,H>,(~L)Iq)}

Let

and

rdLEIS(~,~%).

and

C_Id.

and

Let this

K d HO{i}.

~rdLV).

and

be such that

and

q

rUHCL

[c_w ~"

q~c(F)x H

let the f u n c t i o n

: iEL>

nEH

kok=W11d

Then

Let this

Notation:

Let

: {hELy

z~0.

for some

qez.

and

and

as in 4.7.1.1.

d rdL~.:~ "

zEJ,

J { Ig ( @ ) { y } .

zNc (F) XK#Od @ : L IId

rb L = < < (fi, (~--L)If) then

is

such that

i6~NL

Let

zCc(F)y_

c(F ) .

Lc_~ Let

Let

defined

Then

{)[ d @~(@@V) {y}.

and

is regular.

for e v e r y

z#O

additivity

: He1}

6.2.

225

mE~.~{n}.

By

iEK~F

we have

i P<m,K) @ z

by

arbitrarily

we have proved

QED(Claim

z~c(F)y.

A = Sg{y}

and

U = •215

y

iEAz.

AZDa~L

Since

and hence

pEz

but

ie~L

was chosen

la~Azl

N ILl < w.

/A

for some

satisfies Suppose that

~

and

Then

is regular

~ { rlw'%~

]YI~2

IYI~2

@ Dind nCs

D Cs reg.

QED(Theorem

6.1.)

6.2(i)-(ii)

U@Subb(~). Then

: nC~} E Z d Z ~ { O , ~ U } .

since

Y 5 x•

and since

Let

~ ~ %Z[(~U)~ .

(i) is a c o n s e q u e n c e

Let

Proposition

(i)-(iii).

Let

(< n'H>)

rlwEHO~

rlwEHO~.

satisfies

H c ~. --w

(ii).

Y ~ ] ~ Subb(~).

and 1.3.6 we have that

since

is regular.

C(H)X H = U{~(x•

: tEy},

Hence

4.7.2.1

Next we show that

and

Then

_

6.1.1.)

By Claim 6.1.i,

and

PSm,K> q c(r)y._

and

of

E Cs~.

Then

W~I~ W=~Y

~ ~ Co-dol=l

zq{O,W),

H

= yn~u E R

We have seen that

(ii). Let

z { C~H) . ~(Why)

we have

x

Then

by Then

~[)z=O. -

,

Wr

for some

@i~ Co-do1 =I z=O{aY ([)

Thus

:

rlw*~

below was quoted in [HMTI]6.8(3), (ii) and in

[HMTI]6.10. Propositio ~ 6.2. (i)

CS nLf

(ii)

csregnDc

Let

•

and

~ P Dind

.

~ P Ws

(iii)

H(xcs~egNDca)

(iv)

M(xcsregnDc~ e) _~ P Csa

(v)

H(•

(vi)

H ~Ws

NDc

.

--~ P GwsC~

reg if

x<e.

) _~ p GwsCOmp~ reg _~ P~Cs reg.

(vii)

C S ~ eg

Proof.

Let

c ~

saw.

~ P WS a.

~_>2 and

be an atomless

~_>w. Let BA.

~ d ~[e•

(such a

o~

Proof of

exists.)

Let

(i) : ~

Let

d ~(~

~ c )B.

226

6.2.1.

Then

~ E Cs NLf

PDind

and

since

Zd~:B

~

by e.g.

is atomic

[HMT32.2.24(iii).

for every

Let

K,L~CA

be such that

~ q

~ ~PDind

In the rest of the proof we shall use the following Fact 6.2.1.

Now

fact.

Dind nK ~ LUloCS

. Then

K

PL.

Proof.

Let

and hence

~EDind ~L

Proof of

~loCS ~.

implies

(ii):

Let

Then

~ @PL. HC~

~

is directly

QED(Fact

indecomposable

6.2.1.)

be such that

IHIAI~HI~w.

Let

x

/J~ ~ ~ ( ~

){Xo,Xl].

--

{q~ax

: (ViEH) qi=n),

Excs~egADc

by 1.3 and

(Vi,jEH)XoUXl~dij

IAI>I

and

and

is a corollary respectively,

Theorem 6.3.

Let

(iii),(v)

Let

~PWs

a

and

: HnlfCo},_

XomXl=O

by Fact

6.2.1

since

Let

and

(vii)

of 5.7(iii),(iv) ; and follow from

(v) and

(iv)

(ii)

~Ae.

~.

Then some weakly

is not subdirectly

eZw.

Let

indecomposable.

(H n : new) E w(Sbe)

subsets

of

a

subdirectly

such

that

be a system of laNU{H

n

: nEw}l~w.

we let

G d {x n : new}

Q d ex(O)

(vi)

and

infinite

Xn d: (fe~x

Thus

are corollaries

csregNDc

xZ2

disjoint

x~2

new

of 4.7.2.

Now

since

theorem was quoted in [HMTI36.16(7).

For every

Let

~qlWs

Then

6.2.)

indecomposable

A(Xn)=H n.

IHIkw.

by choosing

The following

Let

Let

la,~Hl~w.

of 5.4(iv).

QED(Proposition

mutually

nE~.

cs~eg~Dinda.

By Fact 6.2.1,

Proof.

n

for all

where and

~J~EDc

{i,[d ~ ( ~ @ ~ ) G . by

We show that zEIg(~)L)G,

~ :d e•

For every

I~NU{H n : nEw}I_>w ~)t ,G

and

Q

z#O. Then there is

new

we

have

and by [HMT]2.1.7.

satisfy ne~l

the condit• such that

6.4.

227

z6Ig(~i){xi i
and

: i
@ --~c ~

: c(@)y.

of 4 . 7 . %

be s u c h that

We show t h a t

: i
pEF~,

@

is s a t i s f i e d .

F --wc ~.

Thus by 4.7.1(I)

~ D m M)

condition

and

S --w c ~

(ViES)@NHi#O. b.)

of 4.7.2 element

of

rlQEIS~

G

subdirectly not

: i
for some

n<~.

By 4.7.2(II I )

Let

(VF --~c ~)y _~ c(r)Xm.

Hence

M~,

qEQqy, (ii)

for some

z~Ig(/)L) (G~ r --wE ~

and

be such t h a t Thus

subdirectly

condition

and e v e r y

y#O.

Let

Then Then

m>n.

Ws

is w e a k l y

that

~A

yESg(Y0~){xi y~Dm N

Then

4] is s u b d i r e c t l y

since e v e r y

i n d e c o m p o s a b l e since

N e x t we prove

N ~ U{H. : i
y @ I g ( ~ ) (G~DmN).

Let

iff

is r e g u l a r by 4.7.2(I)

y6A,

~.

seen that all the c o n d i t i o n s

[HMTI]6.13. Let

z~

zNO#O._ We have

for some

~ ~ rlQ*~ ~Ws

indecomposable.

F c

seen that

i.e.

[ : ~•

/JL is w e a k l y

by

Then

z~c(F)Y#O

Let

x.~ =

Clearly

We have

Let

conditions

and

@ --w c ~(MO~UF)

By this we h a v e

thus

the

q ~ ~[@/f].

where

~

d ~{y}.

f~y

w e have

Let

Therefore

indecomposable

subdirectly

Let

is s a t i s f i e d .

(Vq)q[@/~]qz

by 4 . 7 . 2 ( I I ) ,

~

Then

(Vi
z ~ c ( F ) ~ { x i : i~S}

is regular.

j=i].

zNc(F) (QNy)#O,

(VIES) I H i ~ M I ~ .

are s a t i s f i e d .

Let

qCF/p]Ey.

we h a v e

is s a t i s f i e d .

Let

and

zEIg ( ~ ) [ y }

By

Then

Then

of 4 . 7 . 2

iff

a.) of 4.7.2

~ -wE ~.

such that

@NF:o

iff

QC~y s a t i s f y

l{y}l:l.

exists.

fEF/p]Ey

of 4.7.1

seen that

since

and

d y : O{x i : i
Let

(Vj
be such that

(Hin@)~f~O3 . Such a

and for any

: i
1

~(~ , { y }

(i) is s a t i s f i e d

@ ~w U { H i iff

z~Sg(fA){x

Sg(~l ) {x i : i
Thus

~Ig ('~5){yj.

and

by

is :

[HMT]

XmEG~mm N

decomposable

and by


Remark

6.4.

Cs r e g

is

Let

directly

we k n o w that

NH( c s r e g n D c by 4.13.

6.3.)

~2

and

saw.

indecomposable,

M x C s ~ eg ~ D i n d

) having

It r e m a i n s

I (a~)2f

By [ H M T I 3 6 . 1 5 ,

in

short

csregcDind

But we k n o w more:

-many different

open whether

there

is

every n o n t r i v i a l

There

direct

. is

factor

23L ~H cs reg M

By 4 . 1 3 ~Lf

~

congruences

such that

228

6.5.

~ Ps

with

following.

~ EI(ci~loCS)

Let

~Lf~,

EI(cA ~loCS

).

nondiscrete discrete

~ > Co-dol:l

Then

CA -s

III

and

IIl<w

is an

~ I~1,

because of the

and suppose

~ ~ P~

since no infinite direct p r o d u c t of

Lf

and no m e m b e r of

Note that the algebra

~ELf

(H~)~loCS ~

AM( c s r e g N D c

is

~ c o n s t r u c t e d in

4.13 is such

that

Remark 6.5.

(i) below is a g e n e r a l i z a t i o n of 4.16(i).

~ ~ Co-dol=l.

i n t e r e s t i n g to replace the c o n d i t i o n general but

with

"abstract" condition.

"simple"

However,

It could be

in (i) b e l o w with a more

by

(ii), the most obvious

c a n d i d a t e for this does not work. Let

0 < • < ~ < e.

(i)

(ii)

(iii)

Let

~

Ipl<2

I~

Then

E P{~E

(i)-(iii) :

~[ is simple}.

There is a s u b d i r e c t l y

indecomposable

~+i~

{

j Cs

p-<•

iff

ms

Proof.

Let

for every

for every

~

i
E P(Subu(~)) ]~i ~

~

E P{fAE Cs

~P(xcs

)

since By

= ICse.

P~{EICs~.

by •

and let

{(O : i<~>}.

Thus 6~

Assume

Thus

P#.][ (~ I CS

•

we let

Then

and

~i

~ i ).

P~

Ipl : p>le2[

p r o v i n g that

(i) is shown.

If

P~ •

#J[e Cs reg.

in

P/A

Let

P~

~8~x

p>•

V E

El Gws c~ I Gws c~ ~

.

Pf5 @ ICs

Thus

=

EP( Cs ).

(ii) is obvious.

Let

g e n e r a t e d by the element

There are

by

Let

and let

<'HMTI]6.16(6) ~+i

(Vn_<~)(ViE~)Xn<doi

(~ I CS

is simple

~i d ~(Vi)&~i~ p~

By the proof of

such that

and hence

@ I Cs then

be the s u b a l g e b r a of

By 1.3,

Xo,...,x

Ipl_<2I~I

i
is simple.

is s u b d i r e c t l y indecomposable. atoms

such that

~A is s u b d i r e c t l y

(i) in the proof of EHMTI]7.17 we have

IPBI_>2P>I (e~)21 ~<~<e.

~]~E Cs reg

:

(Vi
For every

~i

by EHMTI]6.2.

Then

iff

p/~ E I C s

.

be such that

be one-one.

Thus

P~5~ICs

We may suppose that

~i

Then

.

i n d e c o m p o s a b l e } we have

Then

below hold.

•

~[

disjoint in

P6[ .

(ii) is proved.

6.6.

229

Now

(ii)

together

QED(Remark

Problems

with

[HMTI]7.29

6.6.

Let

Is

(ii)

How Figure it

Gs

I

=

that

Cs

?

(See the p r o o f

[HMTI]6.9 K Ca C -r s there

let

is

will

look

that

occur

~

~(~)At~a~ Cs

(iv)

Is

HP CS reg = H c s r e g ?

(v)

Find

(iii).

an a b s t r a c t (or

I Cs

replace

K

Along

that

)

with

K

these

lines

@

.

2~

iCs

in we Indeed

1.3.

i GwsCOmp

reg?

HPCsreg

characterization

of

ICs reg

as a s u b c l a s s

of

).

weakly

by

use

[HMTI]6.8(5)

_c H P c s ~ e g ?

Is e v e r y

Is

of

there? such

and

IS

Gs

if we

e Cs reg

(iii)

(vii)

proves

2~<~. HP

for all

note

(vi)

[HMTI]6.12

6.5.)

(i)

I

and

subdirectly

: HPGwsCOmp

indecomposable

reg?

(Note

that

Gws

(or

PGwsCOmp

Cs

) an

reg # P c s r e g

5.6.)

7. ~ r o d u c t s

Throughout

this

section,

Definition

7.0.

(p.l15

algebras.

K d { 6~

Uf

Up'K

That

K.

Uf K of

K,

of

denotes

a cardinal.

Let

[HMT])

K

be a class

of

similar

we d e f i n e

(KNUp{~])

:

d U{Up{%[}

is,

members of

Then

•

and

: /)ICE].

is the and

# O}

class

Up'K

of all

"ultraroots"

is the c l a s s

of all

or

"ultra-factors"

ultrapowers

of

of m e m b e r s

230

7. I.

Theorem the

7.1.

Let

greatest

(It e x i s t s (i)

(ii)

regular by

: SUp{~(w)

I H Gs

: SUp{~(~)}

(iii)

SUpWs~

(iv)

SUp'

:

O.3.9(vi)

} = SUp ( c s ~ e g ~ L f ~ )

~(~)

subalgebra

) = SUpCse

: SUp'lCs

prove

Thm

and

we

have

P{~

since

= SUp'

7.1 w e

P{~(~)}

6.

Let

F

: FCA}EF.

Let

g

~

denote of

~ez.

~A~.

~w

and

: HSUpWs ~#i

=

or

IAIS2].

I o C S ~.

for

all

infinite

~,l

shall

need

the

following

lemmas.

for

Let

~

} c $Up{@~

}.

every

~_>2.

~ ~(~).

By

Therefore

it is e n o u g h

is a c l o s u r e

be

every

Cs

C_ S U p { ~ ( H ) }

SUp

for

: HSUpCs

~Lf

HA2

I { Sb

let

~6w~2.

= SUp '(Ws

Let

Let

if

characteristic

To

c $Up{~}

~

finite-dimensional

has

Cs

@~

cardinal

~

7.1.i.

Proof.

every

: SUp' (csregALf)

: I {~eGs

Lemma

locally

I ~Cs~

caw.

For

EHMTI34.1.)

SUpCs

Let

~w.

operator

an ultrafilter

on

[HMT]O.3.72(i)

by

I

to

and show

[HMT]O.3.70(i).

such

that

(VFeI)

: I ~ Rgs

be

w

{AeI

that

FAg(F):O,

FCI~{O}. ziF

~

Let

{q~

f(F)er FCI.

ziF

and

For

for

every

z.i ~ < zir

Claim

i

iE~F.

: FEI) "

(i)

~{z •

: i<~}

(ii)

z..z. : O i 3

(iii)

~{zi-x i

: i<~}

~{zi'x i

: i<~)

(iv)

zi/F

(v)

A(ziF)NF

# O

: 1 for

and = O

: I ~ ~

iff

we

s

is o n e - t o - o n e

for

define

(ViC~)ZiF~R.

we

have

defined

Let

iE~.

zC~(IR)

(~) i<j<~.

exists

in

I~

for

: (~{Zir.Xir

: i~r)

A(zi/F)

for

for

every

and

this

every

such

j:s(r)i]}

Clearly, By

and

: F >-- g(r)

iEF~{f(F) ]

: iEF~{f(F)J]

every

have

f

s(r)

: (Vj~g(F))[qj=O

Zf(F) F ~ ~{ziF d = O

: I ~ I,

every

= O iE~

every : rci>.

every

and

FEI.

ic~.

x~(IR)

and

Thus

we

7. i . 1 .

231

Proof.

Let

icr,

and

that

(i)

~{ziF and

-one,

and

facts

prove

{FEI

FEI.

By

the

: iEF}

(iii)

construction

: ~H.

This

hold. A l s o

A(ziF)NF (ii),

= O

A(ZiF)

A(zi/F)=O

z

implies

zir. Z j F = O

by

(v).

of

we

have

that

zi/F

~ O

that

for

i#j

~ g(F)

and

follows

from

since r~g(r)

(v)

ziF#O and

also

s(F) = O.

since

iff

is o n e These

(Vi<~)

: i~F}eF.

QED(Claim

i)

Notation:

Define

H ~
the

following

h d ( )~.{z..x. • i E d F*oe

Claim

: FEI)

2

E E

:

: y~R).

Ism(e~

to

ciE(Y ) = E(ciY ) . : Fel>

: F~I

and

Thus

c.E(y) l

2.

Case

: ~{zi'dkn

of

Then

: i<~>

e d hod E

:

yeaR),

and

: ~R - IR/F.

cylindrifications:

:
iCF)

H.

, I]%/F).

i. C a s e

: jeF}

set

d d ((Yii

x E ~ ( I R ) ),

Proof. show

every

mappings:

: i<~}

= (e(y)/F

for

By u s i n g : jeF}

E < ~{zjF.ciY

: cie(Y)/F

Let Claim

: FeI

j

i<~

i,

and

: jeF}

and

yEaR.

cie(Y)

have

= ci( ~ { Z j F ' Y j

i~F)o<~{ZjF'ciY

: FEI)/F

We

:

j : jEF}

: e(ciY)/F

:

= E(ciY)-

= E(ciY)(~)

of

diagonal

(I9%)

elements:

: i<~]]F

:

(dkn

Let

k,nE~.

(I%%).L[z

E(dkn

: i<~}) /F

i

) =

: dkn

(I]%) /~ :

( I t ~ /F) = dkn 3. C a s e

of

Boolean

dEHom(~,

~(I~))

CHom(~4s

~(I]%), ~I]%

antichain exists

in

algebra it

follows

completes We

have

and

in

~

~

for

theory

the

).

~

that

every

Recall

~*EHom(I~

Let

such

that

from

operations:

yE~B.

~{z i

and

By

: i<~}

Then

it

: i<~}

O.3.6(i)).

E : F*ohod.

, I~/F).

d ~{~I~% .

h d (~{zi.y i

EHMT]2.4.7

that

Claim

= 1~<4

is

We i,

Now

by b a s i c

= e~541~=

hE

is an

~{zi-y i

yEeB)EHom(~ ~

that

zE~B

and

immediate :

show

Clearly,

: i<~] Boolean

,~)

(e.g.

~(I~)

proof.

seen

that

E~Hom(~%%

,I~/F).

Let

yEaR,

y # (O

: i<~).

232

7. I. 2.

Then

(3i<~)Yi#O.

~ e c s r e g n L f e ~" = zi.yq .

Let

By C l a i m

Then

by

: c(A) ( z i / F ' ~ i / F ) QED(Claim

Claim

By

Gs

First

z..e(Y)l

we have

and

C(A)Yi:l

= zi'~{zj'Y~

c(A) ( z i / F - E ( y ) )

: zi/F

by

free ~.

formula Then

B=Sg (~)A. H ~ {iE~

c S Up{~(~]

we p r o v e

# O.

I~/FeUp{

Thus

by

: J<~}

=

= c(A)(zi.e(y)/F ) =

E(y)

# O.

~6 }.

p

is

Js

Gs NLf

or

d

. l]

by

H c

such Gs 8

occurs

in

Let of

that

~

CA -s.

that

~]}.

be any

JJLP ~-

contains

be o n e - o n e and

).

such ~

~

~2.

language

since

p : e >--~ L

exists

every

JOt e Gs

~4 ~ ~

: (~j)[c. l

a

for

_c SUp(

is

there

Let

}

in the d i s c o u r s e

Clearly,

(8~e)UH.

•

there

[HMTI]8.5-8.6,

Such

A(zi/F):O

l~l<w

7.1.1.)

Proof.

~

we h a v e

~'R. eSUp{ ~ }

7.1.2.

xGs

Then

2)

QED(Lemma

tifier

l(ii)

: zi/F'c(A)Yi/F

2 implies

Lemma

A ~ A ( ~ )Yi"

~

Assume Let

~ ~

8=e+e.

~

and

no q u a n t i f i e r .

Then

and o n t o

ILI:I~I2~.

quan-

IHl<w. such

Let

that

Then

Let L ~

Hip

~(P)~

c Id.

~

--W

and

~P)~

~(P)~

e Lf~

e I xGs ~.

SUp(

Gs NLf

by

We

)

by

B:SgA.

have

seen

and by

QED(Lemma

Gs nLf

and

and ~ ~.

O.3.70(i).

) _9 S P { ~ ( ~ )

[HMTI]8.1

we h a v e

Therefore

Now,

Gs

C

xGs ~Lf

) 9 S~p{~(~)

},

by

EHMTI]

7.1.1.

return

~_>~_>~.

d_ {< F,~) that

Cl G s B

7.1.2.)

N o w we Let

3.15,

~

that

[HMT]O.3.83

_C SP ( Ws nLf ~ ) _c SP ( C s ~ e g ~ L f 6.2,

By

to the p r o o f We

show

: 2CFC_ -w ~'

{< A, ~ > e I

d ( F,~>eI.

/~ eS~

: FC_A, ~

Then

that

IBl_<~

Therefore

L5 El IGSF

o~ c ~ F ~ [ i

for some

by

of T h e o r e m ~

d

"~?'~}.

~(~) Let

c_ ~L~F ~ }eF and

~5 el

since

e SUp~Gs F

.

Let

I d

be an u l t r a f i l t e r

for

every

Gs F

by

[HMTI]3.18(iv),

/if' E x G s

7.1.

( F,/W>cI.

[HMTI]8.2

~>i_>~

and by

IGSF _9 SRd F I G s

on Let

and by IF1<w.

I such i d ~>_w. Then

by [HFfPI]8.5, 8.7. Now

7.2.

233

~

I ~ PieI ~ i / F

can be seen

similarly

to the proofs

of

[HMT30.3.71,

0.5.15. Then

~(~)ESUp{

SUp{~(~)

} :

_c I (u{ G s C_ SUp

SUp{ ~ ( I ) }

= I coC s

l_>w.

EHMT~2.4.64,

for every

imply

If

(iv)

Clearly,

Cs

:

c K.

I GS

from

by Lemma

is easy

V

0~ has c h a r a c t e r i s t i c

For every

~E~

let

a ~

a :I)

: O<~<w}

to see by

tions

of e q u a t i o n s

then

HSUpK=K.

: ~w}

only,

7.1.2,

[HMTI]7.13 the p r o o f

and

Corollary

Ax

is p r e s e r v e d

CS WI (O{ GS

7.1(i)

and

for and

7.1.2. (i) of T h e o r e m

or

IAI_<2}.

d c (~) d(~• Then

consists under

Let

I iCs

= I {~(i)}. )Ul oCs

,

Ax d

K = Md(Ax)nl G s

of u n i v e r s a l HSUp.

disjunc-

By [HMTI]7.16

: ~Ew--2})Ol i C s

IcsregnLf~ ~ : (I Ws nLf

of T h e o r e m

QED(Theorem

(i)

Theorem

) _c

by I H M T I ] 7 . 1 6

_c SUpCs~ _C HSUpCs~ _C HSUpK:K

_C SUp' ( c s ~ e g N L f )

Lemma

~ SUp'{~(x) by

This,

7.2.

Let

Let

~>i

together

see 3.15,

with

completes

7.1.

for

Lf~ASUpCs~

Proof:

(i) follows

follows

QEm(qorgllary

such that

KE{~Ws~,

(ii)

maw.

be a cardinal.

xCs~ eg,

f r o m 4.15. 7.2)

PK C SUpK

~(~)ESUpK. xCs

= Ud (I C s a n L f

from

Then

)

, ~Ws Nmf but

,

cs~egNLf

SUpCsa

(1) in the proof

for e v e r y

In p a r t i c u l a r

of

K C

PK ~ SUpK , ~Cs~nLf~}"

~ Udl Cs~.

[HMTI37.17

:

[HMTI]7.21,

7.1.)

Gws~ ~

(ii)

K ~ I

NOW

Since

Ax

C_

_c S U p ( l c s r e g n L f

~#i

u {Vx(x<_a 2 V x:l) }.

[HMT]2.4.63.

I Cs

(i).

-

d { (a :O

Therefore

is a v a r i e t y

= SUp{ ~ ( ~ ) }

follows

K d I {~EGs

~,~_>w.

then

= SUp{ ~ (~)} = S U p ( I C s regnLf~ )

then

I•

By [ H M T I ] 7 . 2 5 ( i i )

: }i_>~} _c SUp{ ~ ( l ) }

I ~Cs

~E~2

hence

7.i is proved. Let

by 7.1.2.

: ~>_~}) _c S U p { ~ ( x )

Cs

every

~(~) }

and f r o m

7.1.

234

7.3.

Let

Theorem

7.3.

(i)

SUpCs reg

(ii)

For O

any

l<~<w-<~ 9

# UfUpCs reg nondiscrete

statements

a.

~

SUpg~,

CA

a.-c.

Uf UpGws c ~

which

9

is

not

of

characteristic

hold.

reg

0c

~_ Uf U p D i n d

b.

SUp ~

c.

SUp 6~ ~_ Uf Up (Ws

(iii)

Cs reg

(iv)

H Ws

~

H Ws

{ Uf Up D i n d

prove

# SUpWs

6{

below

--

To

r UfUpWs

. U C s reg) .

_~ Uf Up ( W S a m o C S ) Uf U p W s a

Theorem

and

HCs reg

~Cs~

_~ Uf U p D i n d a

"

~ Uf U p C s r e g

.

7.3

we

shall

use

the

following

definitions

and

lemmas,

Definition a

7.3.1.

denotes

at(x)

denotes

supat(y) ~

the

is

Note

is the

that

Let term

the the

c(~)d(~x~).

formula

formula

formula for

~<w~.

(Vy[y~x

(y=O

Vz[Vx(at(x)

(a ~1

any

~

yeA,

V y:x)3

~ x~z)

~

Vy[supat(y)

~ECA

we

--

-

have

A x#O).

y~z].

(y:O

V y=l)]).

~

supat(y)

~

iff

y =

: ~ it~.

Lemma To

7.3.2.

Dind

~

{%~

Lemma

7. 3 . 2

we

shall

7.3,2,1.

Let

prove

Lemma (i)

Let

x~At~

I{zeA

be

Suppose

A(E

Proof.

such

: z~c(F)x}l<w

= ~{zEht~ (ii)

~w

: x<w}. need

the

and

~CA

that

x-a

for

every

following

lemma.

. r

for

some

F --we~,

~.

Hence

Then

c(F)x

=

: zSc(F)x}. 1~

:

~{-a

x

:

~<w}

and

~ At~

exists

of

It

in

.UL.

Then

A t ~)t ) =O.

Let

ek~

and

~ECA

.

Proof

(i) :

suffices

to

take

7.3.2.1.

235

2<~<~

and

eAtS.

Then,

is

finite

then

we

and

there

distinct all

prove

(i)

for

<~

atoms

non-zero

jo,...,j

s,t~.

~ { ~(-a)~

claim,

has

exist

for

<~

Let

each (so

that

ciY

(i)

pairwise

is an

the

follows

from ]

f3~ . L e t

algebra

easily). Zo,...,z

i

and

of

Boolean

disjoint

{jo,...,j~_l

w : ff{ci(c(F)Zs-di,js) Note

i<~

different

F =

in p l a c e

such

that

ye

~(ciY)~[~

Suppose _1

not:

~ciY.

y~djs,j t

Choose for

set

: s<~}.

{i}-atom,

and

hence

c zl s = c i Y

for

all

s<~.

Now

C(F)W =

ff{CjsCi(C(F)Zs-di,js)

= K { c i c ( F ) zs so

Now

w#O.

and

j E0

we

let

s,t<~,

that

= zt.c(F)Zs.d

:

k y,

s#t.

Let

A:{Jo,...,j

(c(@Nij~})Zs'dj~,j~)'djn,j•

C ( F ) Z s - d A = Z s . d A = z s.

A = O,

so w e

infer

in

= O,

C(FN{jt})zt'di,js'Ci(C(F)

zs'di,js)

C(F)Zt'di,jt'ci(c(F)Zs-di,js)-djs,j

= ~ At~ x-a

#0.

O~F

zt.c(F)Zs

= O,

t = O,

t = O,

3s ,j t"

w ~ d(Fxr):O,

Proof

for

A = c(@~{j~))Zs'd

Hence

ci(c(F)zt'di,jt)'ci(c(F)Zs-di,js)-djs,j

Thus

Now

succession

C(F~{jt})zt.c(F)Zs.di,js

w _< - d

}.

have

c ( @ ) Z s . d A : cj

It f o l l o w s

: s<•

: s<~}

of

(ii): exists

Thus

cix

a contradiction. Let in

I ~

: ~{zEAt~

= ~{-a Let

ic~

: ~<~} and

: zScix}

and xEAt~.

by

(i).

suppose

that

Then

(9•

Thus

c.x~y. l

y d:

Now

A.

236

7.3.3.

ciY

= ~{cix

QED(Lemma

We If

: xEAt~}

a ~ =1

to the p r o o f

then

then

~3~ b

a M :O,

Y = E Ate,

Then

QED(Lemma

Lemma

= Sg{x,y},

~

k supat(y). ~

base(~

~

).

: {b v

: veL}. that

eI Gws

have

the

x~At~.

Let

Ay=O

H ~
there

that

~]~ E

#j%~ s u p a t ( y ) .

Hence

~>1.

y~l,

there

Assume

.~]~.bC~-

Let

x#O.

B =

Then

and

O
and

zCB.

bwEZ.

induces

and

~(y)=y.

ting

=

f-b w ~

Ay:O

and

it is a t o m l e s s

~

~ ~n

Then

satisfy

[HMT31.10.5(ii).

At~

= Ate.

xCQC _ _y the

characteristic for

and

conditions ~

some

and

of

Ax=~.

VeL.

Let

4.7. Let

there U

and

By

x~

and

that

~(i

x =

1

in

is

Then

,

hence

Let

Let

Q-wsmall

and

~ Then

~

~ .

since

: qi~bv].

contradic-

xEAt~

~ Ate=

rlQEIS~

~ ),

~(z)=z

: z~y}.

x

of

bv,bw6x~y

~(-y)~EMn

At~C{z

~ ~Q~.

that

[ U ~ b a s e ( v U w ) 31{

~Els(

then

that

that

F ~ {i~

U

be a p e r m u t a t i o n

f(bw)=b v

proves

Thus

Let

y=UH.

are

f(bv)=b w

R : Sg{x},

~

that

we m a y

v,wESubu(~)

Hence

A ( ~ )Q:O.

H.

that

Then

have

We p r o v e

show

such

zeSg{x,y}

we

set

automorphism

This

We

is of

b E P v E L base(v)

and

Since

B : Sg{x,y}

such

fof~Id,

f(bv)=b w

~

: I Gws w d

every

f : U >~

a base

b w c z.

and

x~y,

by

Then

z

Let

= base(w),

f

for

Since

~
IGws

H~Subu(~)

L~H

f'base(v)

[HMTI~3.1

By

is

and

7.3.3.

for e v e r y

: i<~>

is

~(x)=x

qeV

of

show

we have

Then

hypotheses

we

By

x,Q

Ay:O,

First

~(1~).

-

such

~<~.

by

t

by 7 . 3 . 2 . 1 ( i i ) .

i,jc~,

a~ ~i n

of

Q ~ UL.

be

and

a ~ EZd~

characteristic

have

bvEX-Z

))

~y~.

~EDind

Since

yEA

yeZd~)L = {O,1}

we

i,jE~

that

By

By

Let

~>0,

By

for

bv

Let

a ~M #1.

:I.

for e v e r y

C Gws wd.

~Y.dij

~(1 ~

7.3.2.

Assume -a M

satisfy

characteristic

~Id.

~ .

hence

Let

Let

such

= y.

•

Proof.

U

of L e m m a

i.e.

x~y.dij

for e v e r y

such

ciY

7.3.2.)

7.3.3.

assume

proves

7.3.2.1.)

return

EDind

~ y

~ :Q.

is of Let

IFI<w

qEQ. since

7.3.4.

V

237

is a

Ws

-unit and

bveV,

= U{c(F)x. : F --wc ~}. Thus 7.3.2.1(i) then b

Lemma Proof.

Let

{x,y}.

Then Cs

: new}.

~n

W {n

Then

rlo~Is~ At~

for

Cs

x ~ {0},

since

I{~lq

~O}eI.

Thus

hence

x-dijEI

y ~ {q~V

Then

=

and

xcQ._ By

= Ate.

I.e.

n>z.

~

Cn

y ~ ~(0)

~,x,y

x~y.

in

~

~/I.

A(y/I)=O

Then

and

H~Wsa

~

and

~, ~ ~ a ~ )

satisfy the conditions

by

proving

of 7.3.3.

: qn~O}=2m

~ ~V

and

Then Let

x/mSdij

and

Then

Let

~n"

of 7.3.3.

by EHMTI37.15 ~

Let

~

~

x~I,

Then

i,jca and

~n

mew}, I ~

-y~I.

Hence

~

: ~(i+1)lq~

Ix-dijl~2

(in

and

~/I).

Let

x/I, y/I

for every

x =d

){x,y}.

ciY--y ~ {qEV

i,jea.

for all

Thus

for some

4J~ ~ ~ ( ~

IEII~

iCw.

~<w.

~ el Gws

the hypotheses

: U{nEw

: qEz}I<W}.

O<x/I~y/I
Ws

~

by

l~

~n"

{~n

:

l<~
~)~

V ~ ax(O),

CH

y = ~ At~

then

0 ~ <0 : i
~

~j~/I.

O#x~y#l

Let

Let

{zEA

This proves

7.3.3.)

7.3.4.

Thus

By

qec(F)x.

y = ~ {c(,T,~x : Fc_~}, by

y=~(~) = ~ AtR, thus

supat(y).

QED(Lemma

thus

n>~.

~ ~

satisfy

Clearly

~ e

9

QED (Lemma 7.3.4.) Lemma 7.3.5. (i)

Let

~w.

I~j~/F ~ Dind

be any nondiscrete

for any u l t r a f i l t e r

F

which

CA

.

is not

I~I + -

complete. (ii)

Assume

Proof.

(3~<~) 2j~ ~

Let

~J[

1

mew.

Since

such that

Then

be any nondiscrete

-complete u l t r a f i l t e r and

a r

F

on

I.

is not

Let

yEIA.

Since

~<~

be arbitrary.

~ Then

CA .

~=l+m

: O.

Let

is nondiscrete, {iCl

~ UfUpDind

.

Let

F

be any non-I~l +-

where

~

is a limit ordinal

1~l+-complete,

(I / hlh-l)nF

Then

SUp ~

there is a function

h : I

y ~ < dh(i),h(i)+ I : iEI). we have that

: c yi#y i] : {iEl

O
: h(i)=~

Let

or h(i)+l=~}~F

238

7.3.6.

by the p r o p e r t i e s Therefore of

done.

since

I~/F.

Assume

Suppose on

and

in

(ii):

ultrafilter Then

h

A(y/F)=O

Proof are

of

~

I

~ ~Uf U p D i n d

~

that

since

v,r~PFs I (~F)

be

that

by

such

~ i ~

Then

~

If

Let

I ~ Sb ~

and

~

~

~

for

and

: FC~}~F.

and

i.e.

~

~(~

y#l

F

we

be an

Then

~

is of

: 16w}.

Let

y { < d r F , v F : FEI>/F,

){x,y}.

Then

Ay=O

since

{)[ is n o n d i s c r e t e .

i,j~.

Thus

~

by

7.3.2.

Now

~ UfUpDind

.

~ ~ I~Jt/F.

Let

a #~.

Let

then

let

{ (a I#1 ~ al=O)

{

I~j~/F~Dind

01 q U f U p D i n d

( V F ~ I ) V F # r F.

{rF,vF]3.

O # x~y.dij

7.3.3.

.

Dind~

x { < d ( F u { r T , v F } ) : F~I>/F,

Similarly,

a #1.

is n o n d i s c r e t e

l
(VFeI)[i~F

~

JZd I~j~/FI>2,

(VFEI){AEI

characteristic

(~i~)

Thus

eUf U p D i n d

such

A(dh(i),h(i)+l)={h(i),h(i)+l}.

~

[m

for

~SUpOi

since

every

m>n

completes

the proof. QED(Lemma

7.3.5.)

Definition

7.3.6.

We d e f i n e

%

Vx(at(x)

9~

~

VyEy#0

that

Lemma (i)

~<w~.

to be the

is d e f i n e d

Note

Let

~

7.3.7.

~z(at(z)

iff

Let

at(x)

from

7.3.1.

formula

(a #i - %). atomless

or

atomic.

1<~<~. UoCS

),

moreover

cs~eg : . ~ fi '<s~ eg~ ~ ufup(ws~u0cs~)

(ii)

WS

(iii)

Cs reg

Proof.

formula

A zSy)]).

{}[ is e i t h e r

~Cs reg~ _~ U f U p ( W s

{~

the

formula

to be the

~][b %

Recall

~

{~n

: ne~]

~

Let

1<~<~.

#J[ ~

case

~<2

is o b v i o u s .

any a t o m

z

a #1 n

~<~ of

any

we

and

Assume

Gws

of

J0[eWs

have .

b 9Cs reg

Proof

if

By

Ws

and

since

yEA--{O}.

and

~

(ii) : ~

~k2.

%.

It is e n o u g h

then

~ ~ Ws ~

Let

4IrE Ws

IxI<~

because

Since

O ~

{x,y}

to p r o v e

for .

Let

some

~
xEAt~

(Vf,qCz)ker(f)=ker(q) there

are

fEx

and

Ws

~ % The

and for qEy.

239

7.3.7.

Since

~Ws

there

d w = c(F)xny#O. Thus

there

Proof set

is

of

s

By

is a f i n i t e

x+IxI<~

zCAtt%

let

Let

s { <s

Assume

since

for

Vnx=O

and

~E

•

Ws

some

by

IHI~.

i,j~).

By

proving

~0[~ ~ .

By T h m

seen

of

above

~

d a = h(x)

(~'~)

(i) will

~H6~.

Then

VESubu(~

),

(ii)

(]wEU)VNb

: {qEV

(iii)

VNa

= O.

follows

from

Let IEIl~

and

By

is the

that

C s ~ eg

there

and

any

and

is no a t o m

V ~ ~(P)

VAzE{O}UAt~

below

Then

hence

~ CDc

.

By

~>l

(since

either

w~d

have

VNz=O.

A contradiction,

since

x,y~Sm ~

be as in the p r o o f that

from

For

43~ ~ ~.

(ii)

and

by

w~-d

13

9

of

We

or

13

(iii).

show

We h a v e

that

H~c

(iii).

EHMTI]7.15.

Let

hEHo(~J[,~ ),

(Vi,jEH)b_
Then

and

and

(V0c ~ ) a - c ( @ ) b = O .

one of c a s e s

(i)-(iii)

below

holds 9

Vnb:O.

: H1q~w}

and

Vna=O.

(i)-(iii)

holds

for e a c h

VESubu(~

)

(~{).

~/I

~

N = Sg{Wna, A(W)=O

1.3.3

~E

is such

o n e of

W ~ u{VCSubu(~

Clearly since

exactly

Aw:~

%.

: H1q~]

B = Sg{VNy}

~ ~ E Gws~ d

and

= {[}

that

~b

lwl<~.

IHINI~HI~w.

pEz

Then

hence

Then

Then

(~uCU)VNa

fact

Let

should

U = base(V).

= VNb

and

y ~ {q~

show

we

follow

= b d h(y).

(i)

The

1.3

~

that

{0},

We

(e--F)IfCq.

proving

z E A t ~ we h a v e

~ ~ Cs~ eg

(Vi,jC~)a
Let

9

~Jl,H,x,y

Then

and

~ CWs

~eDc

Let

that

c ICs reg. Let

(i) :

z.

(VwCAt~)

for all

Proof

x :

some

By

we h a v e

~

Let

that

Ic(F)xI<~

be such

xEAt~.

for

such

z~w~y,

H C ~

Then

y~zEAt~

rlv~HO({~, ~)

that

: i<~).

[~([~){x,y}. y.

we h a v e

such

(iii):

F C ~

then

greatest

in

) : Vea:Vnb:O}.

~(-W)~ Wnb} i~

.

Let

~

: Sg{O}. we

zESg ( ~ ) { O }

have

~ ~W

of

I ~ ~ .

Hence A (~)z

: Mn(~

Lf-subuniverse

Let

{zEC Let

: z~W}. zEI.

IA(~)zl<e. : A (~)z.

)

since

~

).

If

a,bESm s W=I ~

Then

By

Thus

zEN.

z C _ W and IA ( ~ ) z I < w .

(and then

Clearly

since ~

EMn

Dm~ n Gws

240

7.3.7.

and h e n c e

~ e l C s reg

and we are done.

I~C.

~E

~(~

By

z=O

by

Thus

zEMn(~

rlwCIS~

Let

Gws

,

).

Since

.

).

: H1q~w}

Let

~

with

g(VNb)=y

(**)

zCI

Assume

~ rl(V)*~

.

wEU

Let

since

~

hvy(VNb)

Let

and

R = Sg{VNb].

and

Let

and

and Y#V).

zNy#o].

f(a):O=rl(Y)a.

By a c o m p l e t e l y

analogous

We have p r o v e d

and of

and

I{VESubu(~)

: f(b)nV#O}I_
and

(~VCSubu(~))

(f(a)nV#O

Assume

and

b#O

~

•

= <x,O>

Then

T[ ~ ~

a#O.

A b o v e we p r o v e d

Similarly

~ and

.

there

wCU

VNb =

be fixed.

is

gEIs(Z,~)

, ~ys

))

Moreover

(VfEHo(~))E(fa#O

d ~y~.

Then

such that

, ~y~

)

hvy(Vnz)*O ,

and

f(b)=

f(Ts thus

=

VNz#O. We proved

: Vnb#O}I
we can p r o v e

,~)

0
T (x)

Then

I{VCSubu(~)

Then

(VVESubu(s ].

with

Subu(~)

the e x i s t e n c e

]~ E Gws wd

we h a v e p r o v e d

fbr

~

with

such

below.

~ f*2O[ ~ 6~ 3.

f(a)cV

,{][)

with

h(fa):x.

tEIsm(~[,2~)

a__ ~ ( ~ ) [ •

(~**)

with

g~Ism(~y~[

,~)

and t h e r e is Let

= {V,Y}

of

kCIsm(~v~[

= < O,y).

and

with

f(b)Nv#o) .

exists

I c_ ~•

t(fb)

or

zeC

fEHom(~

fCIs(~

: f(a)NV#O}I_
t(fa)

I={O}.

(*~;) below.

YNa*0})EIss

l{V~Subu(~[)

I c_ ~ V ~

Thus

(ii),

there

is a t e r m

Hence

argument

the e x i s t e n c e

g(fb)=y.

(:{**)

~

i.e.

E V N a # O ~ r l ( - U { Y E S u b u ( ~ 7 ) : Y#V

Let

Then

Let

T=O,

f(b)C_Y.

By

(3hvyeIS(]~Vs

There

f d (hvyorlv).

Hence we may assume

and

By 3.14

and Y n b # O ~

rl(-T)Els~.

i

)=O.

we p r o v e d

Let this

This p r o v e s

= YNz#O.

Case

U:base(V).

VNa=O.

= <~{(rl(Y)a,rl(Y)b) = rl(Y)~s

that

arbitrarily

~ ~(~){y}.

) : Ynb#O

[Ynb#O

z = T ( ~ ) (b,a). =rl(Y)b:Y~b

InMn(~

Then

= Ynb].

T d u{YESubu(~

(gYESubu(~))

W#I ~

W:O.

Vnb#O.

for some

therefore

and h e n c e

was c h o s e n assume

(VV,YeSubu(•))[Vnb#O

Let

is s i m p l e

H e n c e we m a y

VCSubu(~

= {qEV

)

Assume

with

Then

7.4.

241

Since

IdeHo~,

by

(*=~*) we have

Thus in the present Case 2

Assume

above we have Case 3

case

a=O ~

Assume

~el/ACl

and

~ ~

b#O.

_c ~

a#O

(VfeHof)t)[(fa#O#fb)

and

~

e Cs[ eg, b=O.

We have proved have also proved (****)

~ ~ e Cs H~Cl

M

~

Case

i

el ~Cs reg.

We could treat this case analogously By using parts of the above proof one

for some

Cs reg.

~

Then by 1.3

~

is regular.

A c t u a l l y we proved more than this, we

(****) below

That is there are exactly

~[){y},

4 isomorphism

types

@[}. in

H~]L.

7.3.7.)

Now we turn to the proof of Theorem the first part of follow

as it was desired.

hence

HO[ = I { "~tW(23[), [~(~J[){x},

QED(Lemma

~ t%3.

Then by the proof preceding

to Case 2 but we can also do more. easily proves

Cs reg

~ f*~

(iii)

from 7.3.5(ii),

follow from 7.3.7. upon o b s e r v i n g

(iv) and the rest of

7.3.

UfUpCs reg # UfUpWs The rest of

(i), and

Ws u c s r e g o G w s c~

(iii) follow from 7.3.4,

using

and (ii)

regcDind

.

Ws clCs reg.

QED(Theorem 7 . 3 . )

As a contrast and

P Ws CSUpWs

Proposition

7.4.

There are

(i)

to P r o p o s i t i o n if

~>i,

Let

There are ~ x~u

(iii)

Proof. ~

~ ~(Z

~,~e

f UpGwsCOmp r e g

)G

O<~<~e. and

Let

7.2.

such that and

Ws

P cs~egcsupCs r e g _

Then

~ ,~ 6 Cs[ eg

P Cs~ eg ~ UfUpCs reg

Let

by Corollary

O<~<~.

x~UfUpcsreg (ii)

7.4 below we note that

S 2~

~ UfUpcsreg.

such that and and

S 2g~ P Ws

~ UfUpGwsCOmp

U ~ (~+~)~x,

~ ~ I~&(~U).

~ Uf UpGws c~

and

~

By Thm 1.3 then

reg reg

~ [~.

G ~ Ats ,

~,~c~Cs

reg.a

If

242

7

~=i

then

~•

k

~x(O<x
V x I (O<x
~>i.

Let

At~

Ate=

O.

= G,

Def.7.3.1. Lemma

Let

easily proves Both proofs (s

G + ~ At

of

G,

, Let

~@V)G+, ~

Problem

7.5.

What

the

~

O UfUpGws c~

~

~

and

~ ~(W).

a

proof

for

(iii)

see

by

T h e n one as above. s

W ~ ~U ([).

Replacing

(ii).

~n+l'

(i). Let

in the a b o v e proof of (i) by

we o b t a i n

since

reg,

~n+l

the rest of

V ~ a~(O)

Let

every

G +,

~+

and

is an i m m e d i a t e

7.4.)

Let answer

l<~<e~. if

Ws

Let is

~ C Ws

replaced

by

Problems

7.7.

(i)

Is

Uf UpCs

(ii)

Is

Uf Up Ws

(iii)

Is

Uf U p C s r e g

(iv)

IS

Uf Up Cs reg = Uf Up( Ws U o C S

(v)

Is

Uf Up C s r e g

(vi)

Is

Uf Up ( G s r e g N L f a )

(vii)

Is

Uf Up ( D i n d

Let

Proposition

7.8.

Upt.)'t <~ SPDc

.

Proof.

proving

~

b

Assume

: xEG},

: xEG}U{( 1 ~ ,O),}).

proving

~+

.

= {< x,O>

or d i r e c t l y

s.

and

x,O)

D UfUpDind

At~

~ UfUpDind

~x~

~ ~

~

~ SUpDind

reg c D i n d

of the above.

QED(Proposition

is

~x~

~ ~(

GwsCOmp

< 1 ~ ,O> = ~ A t ~

for all sets s

while

Then

~ ~ UfUpDind

respectively,

corollary

~ •

Thus

either

{~V,

occurrence ~+

~=

~ { 9(~x~J~)({<

show

: i<~)

~x~

Thus

This proves

7.3.2.

: dol )

9 5.

Let

.

Is

2)0t6UfUpWs

Cs reg

true?

in

both

places?

,

~km.

Then

1<~<~_<~. = SUpCs

?

= SUp Ws ?

Let

I d Sbm~.

= uf U p G w s c ~

reg?

)?

= I Csa?

nGws

~

= Uf Up ( c s r e g n L f a ) ? )

= Uf U p G w s c O m p r e g ?

be any n o n d i s c r e t e

x d < ~ { d o i : ierNi}

CA

:

tel).

Let

F

be an

7.6.

243

IGws

SUp

Gws c ~

=

HSP (Ws ALf )

=

SUp(csregALf

)

=

HSUp(Ws

/

I

Cs

IGw

= I Ows c O m p

reg

Uf Up Gws c O m p

s/ comp

re

~

/

)

/

/

/

/

/

/

/

SUp (Ws nLf

I oCs(UUf Up Ws ~ Uf Up Ws

/

/

)

/

Uf Up Cs reg

/ / I (oCS UWs ) ~ws

reg

NLf

Uf Up (Ws ALf ) /

icsregALf ~-~Ws

ALf

F i ~ u r 9 7.6.

The

figure

where

=?

know

whether

7.6

appears

gives we

UfUpCs

all v a l i d

do not k n o w = SUpCs

(~Z~)

inclusions about

or not.

the

and

equalities,

equality,

e.g.

except

we do not

244

7.9.

such that

ultrafilter

= (cix)/F = i.

= < cix F : FEI>/F

Clearly

x=13

x/F

we h a v e

Corollary

# 1,

7.9.

Let

F

~e,

T h e n by

Then

~2.

By

without

EHMTIJT.2,

Proposition

concerning

Gws~ ~

: ieF}eF SPDc

~

Then

ci(x/F)

we have

cix/F =

E(A{cix=l

: iE~})

HSPLf

# SPLf

, Up( c s r e g N M n

=

)

I.

The n o t i o n of an

and the f u n c t i o n

associated with

~6Icrs

and

~ ECrs

c

RePc

:

were defined in [}gffI3

(VieI)base( ~i)=U i

.

then

RePce

We shall use these n o t i o n s

is a g e n e r a l i z a t i o n

corollaries

of P r o p . 7 . 1 0

closure

properties

of

[HMTI]7.3-6.

analogous

We omit

to [ H M T I 3 7 . 8 - 1 0 ~ norm G w s ~ a, u w s ,

of the c l a s s e s

etc.

(iii) of P r o p . 7 . 1 0

(among the o n e s

investigated

nontrivial

Let

if

7.10 b e l o w

, Gws a

Proposition

c

set

their definitions.

various

(i) and

function

for some

recalling

the i m m e d i a t e

on some

~ Sb~(PiEIUi/F)

EHo(PiEI/3~i/F, ~ )

I.

EHMT3.

be an u l t r a f i l t e r

: PieI(Sb~Ui)/F

every

by

{FeI

iE~.

7.8)

(F,
By

and s i n c e

Let

.

Let

7.1.

: F~H}EF.

I0~/F ~ SP-Dc .

QED(Proposition

SPDc

(VFEI){HEI

7.10.

below

here)

regularity

of

is the o n l y p r o p e r t y

Gws -s w h i c h

is d e s t r o y e d

under

RePc.

Let

U ~
U,~)

-choice

(i)

Let

~eIcs

Then

RePc*P~/F

~EIGws

) : icI>.

func t i o n .

Then

reg

and let Assume

F

be any u l t r a f i l t e r

PU # O.

(i) - (viii)

Let

GwsC~

iff

(ii)

Let

K e { G w s nOrm,

If

(iii)

Let

K e { G w s wd,

Gws n~

Gws c~

Then

for e v e r y

nonzero

x@PA/F

be any

(F,

b e l o w hold.

be a s y s t e m of n o n d i s c r e t e is r e g u l a r

c

on

F

6~EIK

is

exists

e~e.

IaI + - c o m p l e t e .

then

Gs, Cs, Ws} there

algebras,

RePc~p/Jt/F E K Sand an

~)[EIK 9 (F,U,~)-choice

7. IO.

245

function

e

such

base(RePe*P#O~/F) (iv)

Assume or

that

let

that

F

then

(v)

If

(vi)

Assume

that

there

is an

is

F

is

Let

set

I.

we

show

Suppose Dind

that

F

eXF

such

from

Lemma

be

two

that

one-one

assume

that

the

limit REF

i ~ F '

H

Claim

i.

be

in

ECK].

such

that

since

and

Let

A(z/F)

and

O,[cpu

: X - Sb Let

y+ i ~ F}.

i<8

I

= 0

such

(iii)

and

if

nonzero

that

xCPA/F

RePe(X)

# O

PI]~/F.

be

any

We

F

there

are

RgfnRgt

every

ieI.

(ViEI)2~U i such

that

p

by

the

: jsu}nz

: j~i}nz.l and of

O ~
(VjEX)Yj

is a l i m i t E F

t

such

: X >~

We may : icI)

d= {ieI :

and

u<X>.

because

Let ~p~X

p

by

is a

: i~p >.

RgHnF

We

: (Vn<2)

ordinal,

that

X

hypotheses.

H d= < y+ i ~ Y + i+l R

ZE

follows

t(O)=O.

Let

First

and

easily

and

and

= 0

Let

L~I + - c o m p l e t e .

k=IA]Nc

~

: I >~

-s.

on

RePc*P4g/F

f

that

Cs

that

Let

Y+ i

ultrafilter

is n o t

show

8Gws.

E

function.

This

Then

such

)}

Zi ~ F.

is a p a r t i t i o n

in

every

nondiscrete

if

y+ =d < n { y j

R d= Y + O ~ n i < p

be

in

O}

~i<%

( ~ i < p ) y + i = n{yj

zEPA

as

= PU/F

(F,U,e)-choice

regular

Then

n

be

RePc*P4)~/F

F

of

any

for

: p ~ SbR

( V m < p ) H m ~ Y m.

c

an o r d i n a l

ordinal 9 Let and

a system

is

Y

K

Subb(~i

and

I~l + - c o m p l e t e

j ,n / F ) i = n ] .

P d= n { B E ( X + I ) : ni
is n o t

function

c(fj,n/F)i=c(t

y+

F

functions Ui

~Aw

I~l + - c o m p l e t e 9

on p . 1 8 0

~ ( i : ieI>. define

not

e

then

be

(Vi<j<X)Z.DZ.I_ ]

4.2 9

~IGws

is n o t

is

let

for

: Y~PiEI

Let

Let

RePc*pZ)i/F

Since

s

~EIcs

: icI>.

Then

function

{py/~(U)

(i).

that

.

and

= PU/F.

If

prove

Let

U ~
II~/FI.

and

base(RePc*P~Ji/F)~{PU/F

(F,U,~)-choiee

(viii)

First

# O

.

l~l-regular.

base(RePe*P0i/F)

B ~

RePe(X)

base(RePc*Pt~/F) ~ K

then

Subb(RePc,pffg/F ) !

some

Then

RePc*Pff[/F

(vii)

Proof.

I~I + - c o m p l e t e

~)[ E I G w s c ~

and

,

= PU/F.

K = G w s reg.

4JLCIK

RePe*P~IL/FEK

= 0

Then and

(Vm
246

7.10.1.

Proof.

Clearly,

m
Let

T ~ {iER

is o n e - o n e .

entirely

Let

A(z/F)

C t*p o f*p.

: jEA(zi) }.

Thus

Then

j ~ A(z/F)

analogous.

Let

by

T:H m

be such t h a t

p ~
=O/F

and

q(fm)=~/F].

Then

in p a r t i c u l a r

Claim

2.

peRePc(Z/F)

and

q ~ R e P c ( Z / F ).

Let

The case

i~H m .

= c(fm,O/F) i = O

and

c ( f m , ~ / F ) i = I.

and

+ ( {)~i) (c q)i @ d t m , f m : z i.

then

= zi

pERePc(Z/F )

Let ECs hence

base(~

IZd ~

not r e g u l a r F

~

is not

use L e m m a

Lemma any

by

~

) = PU/F

1>2

7.10.1.

in

by ~

CCs

.

Proof.

is

The p r o o f of

the s t a t e m e n t and r e p l a c e

Suppose

"Let "iEI"

now

RePc*P~]t/F. y = RePc(X/F)

~

by

(Vm
we have

c(tm,O/F) i =

By

U{H m

: m
Then

RePcEHO(~

{ Dind seen

i

To p r o v e

Therefore we have

D Gws~ ~

that

,~) p,qCl

~ and

A (~)Repc(z/F) reg

Thus

RePc*P~/F

the o t h e r

and

~

= O is

is not r e g u l a r

direction,

we shall

below.

~][eIcrs

(VieZ)[x i

RePc(X/F )

is

QED.

By C l a i m

We h a v e

(F,(base( ~ i ) : iEl>,~)

Then

and since

(c+P)i = < c(j,pj) i : j<~)e

EHMTI]7.1-6.

and h e n c e

Let

be such that

Thus

{ RePc{:~ .

lal+-complete.

7.10.1

Hm -C ym

q ~ RePc(Z/F).

Let

O
proving

if

and

~ ~ P2J[/F9 and

for

jCf*p

and

pO=qO

and

( ~i )

T h e n by

: j

m
Edtm,f m

j:tm

RgtnRgf = O

by

H m q F.

Let

QED.

p,qe~(PU/F)

Proof 9

jCt*p.

is

,

F

any u l t r a f i l t e r

-choice

F-regular

F-regular

in

EHMTI]7 9

proves

"iEZ"

~][eIcrs reg We h a v e for some

and

in

~i

the p r e s e n t

with

"assume

throughout

F

is

to s h o w that

~

xCPA.

Let

I

xEPA,

and

and

c

ZeF

A(xi)

be

and

F~

~ ~(x/F)UF].

RePc{~P~/F.

F = lOAf(a/F)" with

f u n c t ion.

on

Let

lemma

all the h y p o t h e s e s "

the proof 9

l~b+-complete. is regular.

A { A(~)y.

if we r e p l a c e

Let

Then

QED(Lemma

Let yER.

7.10.1)

~ Then

A = A(P/Jt/F)(x/F)

7.4-0.

247

since {iEI

RePc

is an i s o m o r p h i s m

: j ~ A({~i)x.} i

-completeness

I~i +

for

of

F

every

and

by

[HMTI37.3(ii).

jEe.

by

Then

Z ~ N{y

A = A(x/F).

We h a v e

Let

Y

]

: jE~A}EF

3

by

(ViEZ)A ( ~ i ) x

C 1--

~A.

Thus

y

is

by

(ViEZ)~i

is

IUA - r e g u l a r

~(~)y=A. For

in

~

We h a v e

the p r o o f s

IUA - r e g u l a r

of

by L e m m a

seen

(ii)

that

-

in

7.10.1.

~

(viii)

4jt~,

by

~iECrs

Then

y

reg.

Then

is r e g u l a r

in

is regular. assume

the

notation

in the p r o o f

of

[HMTI37.4. T9 p r o v e enough

to

show

we

then

have

: Yi,ji=Yi,ki } u

{iEI

is clear.

The

proof

For

(v),

and

The

of

so t h a t

proof

of

x

any

The

the

case

and

[HMTI37.5,

and

the

~IGwsn~

be d e f i n e d

(Vz,wEPJ)[z

Let

= w

P : PJ/s

d = <
C(~,pj

Such

a

c

by the

iff

~ PJ

So

let

(3)

so o u r

and

so

{iEl

(ii)

follows

aEPA

let

conclu-

holds.

therefore

checked

from

from that

rEPJ.

Let

as d e s i r e d .

7.10(ii),(v) 7.10(v)


jEpJ

0 qxe-Wj

that

Ws ~ G w s ~ ~

be such

for some

# O.

7.4(i). We m a y

equivalence

following:

{iel

: Yiz(i)=Yiw(i)

: zEpJ/~>.

function Let

)EF3.

such

c : ~•

that

p(r/~)

: r.

~ PU

be any

choice

Let

that

o~(z)/F)

exists

= pj

by the

o~(z)

following.

for all

Let

~
Z,wEpJ/-

c

directly.

follows from

or

: ji#ki]

desired

X = base(RePc*p4~/F),

follows

it is

QjAQ k = O

Clearly

(viii)

# O,

and

then

it is e a s i l y

K = Cs

be a c h o i c e

: iEI>

such

(i)

PJ

and

K : Gws c ~

(~iEI)Pir(i)eai on

# k/F

is similar,

(vii)

Then

K = Ws

and

(2)

: Yi,jinYi,ki=O},

case

case

By

: ji#ki}eF.

is in the base.

(iii):

that

{iEI

gives

xEX.

[HMTI37.2,

assume

j/F

base(RePc~P~]~/F)

and

Let

and

K = Gws c~

[HMTI37.4

Take

K = Gws n o r m

j,kEpJ

case

suppose

q~Wj.

also,

that

if

Indeed,

{iCl

suppose

that

Qj=Qk"

sion

(ii),

zEPJ/-.

be such

that

z#w.

248

7, { 0.

Then there

is

Z~F

such t h a t

(VicZ)Yip(z)i#Yip(w)i

Yip(z)inYip(w)i:O

by n o r m a l i t y .

(V•

~ (pj o~(W))/F.

(pjx'~(z))/F We s h o w that

Let

c

f { RePc.

is a c h o i c e

Then

by

EHMTI]7.2

since

of

EHMTI]7.2

by

f(V/F) c

and

p(r/z)=r.

Let

y = t/FEX.

Let

and

q ~ < pjxo~(z)/F

Then

a

the d e s i r e d

Gws~~

satisfy

: ~<~)(O/y).

of the last part

(~iEI)tiEYiv(i).

Then clearly

f(a/F)#O

base(f(V/F))=PU/F=X.

to show that

be such that

Thus

properties.

by 7 . 1 0 ( i i ) .

the h y p o t h e s e s

It r e m a i n s

v6pJ

(ViEZ)

(ViEz)(V~<~)~(z)i~#~(w)i~.

function with

is a

and h e n c e

Let

qEf(V/F),

z ~ v/~

and so

y6

Ebase(fV/F). Suppose IYijl>l. above,

(2)

now that

(J[eIGws wd.

Then there

is a c h o i c e

W e may a s s u m e function

c

that

which,

( V i E I ) ( V j E J i) in a d d i t i o n

to the

satisfies

c(~,q) i # P i p ( z ) i ( ~ ) ,

for all

~<e,

# pj•

T h e n it is easy to see t h a t

iEI,

zEPJ/~

and

qEX,

q #

/F.

f(V/F)

is a

Gws wd -unit,

see the p r o o f of

7.14.2. Suppose (i),

that

satisfies

(3)

~J~eIGs

.

Then there

is a

Yip(j/~)i

for all

~<~,

Then it is not h a r d to see t h a t c o n d i t i o n -unit~

see also

P r o o f of that

(iv):

in any case,

it s u f f i c e s

: iEI)/F

function

such that

(c

q)i=Pi,ji

Now

for

(4) in the p r o o f Assume

that

F

of is

base(feP~Jl/F)=X.

to show that for e v e r y

= < Pi,ji

,+

which,

in a d d i t i o n

to

(3) below.

c(~'q)ePieI

Gs

c

for all c'(~,q~)

for all

K = Gws reg,

~<~,

jEpJ

(3) e n s u r e s E HMTI]

qEQj. that

jcpJ and let

we have

: ieI>

is a

F i r s t we c h e c k

PU/F = O{Qj

c'

f(V/F)

7.4.

lal + - c o m p l e t e .

Clearly

= < Pi,ji

and

Wj#O.

be an for all

ieI,

and so

qEWj

using

(iv)

follows

f r o m the above,

: jepJ}, Let

(F,U,~) ~<e.

so

q~ = -choice Then

EHMTI]7.3. f r o m the p r o o f of

7 .'1~..

(i)

249

and

from

from

(iii)

Proof

and

of

-regular

EH M T I ~ 7 . 3 ( i ) . from

(vi):

For

the

remaining

choices

of

K,

(iv)

F

be an

follows

EHMTI]7.3(i).

By

(iv)

ultrafilter

on

we may I.

assume

Then

that

there

ah~.

is

h

Let

: I - Sb ~

such

I~I-

that

w

(V~<~){iEI Let

: ~Eh(i)}eF.

sEPa

PU for

and

and

let

Let

q ~ < (pj

j : X ~ PJ

all

yEX.

c(~,q~)=pj

Let

os

be

the

choice

for all

~<~

Such

a

Let

yeX

w(Y)i

c

exists.

Then <w.

Let

(c+P)i

function

c

a~PA Then

that

Let

w

PASO. : X

w(y)EP
: ~xX - PU

be s u c h

: ieI>

that

and

~eh(i)

for all

f = Rep(F,c).

y~Rgq

If Let

p~f(V/F),

be s u c h q~X.

and

~<~,

Now then

p {
thus

yEX~Rgq,

f(a/F)#O

and

since

y~base(f*P~J~/F)

: ~<~).

= E(a~h(i))IPiyUh(1)lw(Y)i3EaYiy

QED(Proposition

yEbase(feP

Let

iEI.

(Ply)

_c Vi,

i~I.

qef(a/F). since

Let

qE

d y = j(y)i.

since

lh(i) l<

~/F).

7.10)

conditions

formulated

w(y)Ey

if

yEXNRgq.

Therefore

All

that

~{h(i)

be a r b i t r a r y .

Ef(a/F).

: ~<~ >.

if

Let

and

os)/F

such

{pij< )i c(x,y) i =

~EIGws

of P r o p . 7 . 1 0

in R e m a r k

above

7.11 b e l o w

are

which

needed.

Part

is q u o t e d

of this

in the

second

is part

of

[HMTI]7.7.

Remark 1.

7.11.

(Discussion

Statement

(ii)

G w s ~ Q, WS e, GS 2.

Statement let

I

exists on have 3.

The

I

of

7.10

~w

~EGws and e v e r y

of

7.10

not

cannot

be a r b i t r a r y . such

that

generalize

to any of

be g e n e r a l i z e d Assume

for e v e r y

(F,
base(RePceI/j~/F) condition

does

7.10.) the

classes

.

Namely,

, Gws reg.

(iii)

and

of P r o p o s i t i o n

III

<

to

Gws

I~I.

Then

nonprincipal

: i6I),a)

there

ultrafilter

- choice

function

(v).:

any

c

~I(base(f][))/F.

~][EIGws c~

is n e e d e d

in

For

~6

F we

250

7.1 2.

I(Gws ~ G w s ~ ~ is an

that

base(RePe~P~]t/F ) ~

ultrafi!ter

F

base(~)

Under

the

assumptions

which

the

inclusion

conclusions space,

Theorem replaced

the weaker

isomorphism

to

instead

7.12

I,

of

(iii)

(vii)

of

e

on

I

such

finite

base

for a n y

U,

~eGws

for any

[~[-

nl {I~j~/F]

we

there

is a c h o i c e

can be replaced

function

by equality

e and

for the

hold. of the

implies

that

condition

x>a

function

F

~k~.

the p r o o f s

below

-choice

with

on if

in

(iii)

we omit

can be weakened

Theorem

of

7.12(ii)

with

~ IU/F

ultrafilter

{Pielbase(~Ai)/F,O}.

~Gws

have

To s a v e

-complete

]~l +

(Fr,~)

any nondiscrete

-regular

5.

any n o t

there

For

4.

and

in

above

the

~>~

statements

condition

in

1-5.

~ = ~

I~I

EHMTI]7.25(ii).

[HMTI]3.18(iii),

too,

if we

can be

I~I

:

require

only

ext-isomorphism.

Let

and

8

be t w o

cardinals.

Let

~,

8Z~

and

(~CBCs ~ 9 (i)

If

~ ~

IAI +

~I +

(ii)

If

x A

IAI +

~1+

sub-isomorphic (iii) L e t

then + B,

to a

x Z 2 I~uB

~J[Cl Cs or

Cs

if

.

• _> 2 [~uS] ,

then

over

Theorem 7.12(ii)

Lemma

the

Then

every

Let

8Cs

is

7.12(i)-(iii) follows

7.12.1

eSubb(~J[)}). (i) (ii)

GCH.

Gws

~ If

strongly

c I Cs

Let

~Z~,

Then

(i)-(iii)

then

(ii)

~ e Gws

and ~

strongly

.

le[

hold

and

.

.

to

7.12.1 (iii)

Let

below. of

~ Z

for s o m e

(VV~Subu(~))Vnl

~ Ca

BCs

Then

sub-isomorphic

from Lepta

from applying

E Gws c~ ~>a

x ~ B +

follow

is

.

+ (iv) A s s u m e

{~

more-

some

.

together.

IAIN(21~I+~{2

#0.

Cs

In p a r t i c u l a r ,

7.12.1

~,~ClO[ ~

_c I ~Cs~'

.

IYI

:

yc

7 .'12 o& ~

(iii)

25I

Let

HA[base(~)

and

~

and

then

Proof.

Then

rl(~ ~

is s t r o n g l y

~;

~J[ a n d "

Let

~

i.e.

:VESubu(~)).

(][ is s u b - i s o m o r p h i c

) : Subu(~

~

is n o n d i s c r e t e , pEP(V

I.

Subu(~][).

sub-isomorphic

be

as in the

IAI#Z.

Let

) >~

Let

to

~

If

We m a y

Y = (base(V) Then

0t~Gws c~

.

hypotheses.

I d Subu(~J[) ~

to b o t h

assume

that

: VCSubu(~)>

(Vi~I)i

a~ld

: ~y!pi)o

If

A

>- iU{Rgp i -< IAI

iJi

and

_>iU{Rgpi

and

Then

iA] >

Let unit

Then

with

unit

j d I,

Otherwise

J=I

if

I~I

hence

by

and

let J

JCI

be such

is such

that

since:

Suppose

IIi-i~[

_< 2 i~i

+ ~{2 iYii

on

Since

x <

~.

~

that

x _>

~<[U{Rgpi: : icI}.

is n o n d i s c r e t e

}~ -> IJi 9 l ~ i .

: iEJ].

Then

[HMTI]6o2,

~y (pi~" .

Now

~>Ibase(~][) i ,

by the h y p o t h e s e s

Z d u{C~y!pi)l Z.

let

~
x _> IA[ -> IJl

we h a v e

Ws

then

(VaEA~{O}) ( ] j E J ) a N ~ Y ! P J ) # o . 3

: iCJ][

: iEl}i.

Thus

: i~I]]

0[ ~

We may

rlz~ e-Is(~J[, /S) I _c p i e J ~ i

assume

for some

where

iBii

<

IAi

each

~

with

~z

for all

is a

iEJ.

Then

1

[Bi[

< x

each

by

~i

such

that

S

< 2 '~UYi[

is sub-

be

Let

U

a set

By

[HMTI]3olS(ii),

or e x t - i s o m o r p h i c

~:iTii:ITi~R~pi!

isomorphism. Let

iBii

.

be a set

such

that

to a

~LEws{;

Let

riels( ~_~o~i )

such

that

Ucs

and

iS~UJ:~{

=

iSNRgpii ,

( R g p i ) I k i C Id. where

~.

there

This

is of u n i t

induces ~S (pi) o

Let

sub-

>~

l

S

W d u{~s(Pi)

or ext-

BV ~

k i : T.

~T(Pi)l

and

icJ

a base-isomorphism Let

unit

~ iCJ}

"

is a b i j e c t i o n

ki

with

be this

U D <]{Rgpi

--

~Rgpi[

ai]d [ H M T I ] 7 o 2 5 ( i )

!UI<_H. [T.~

-

I

such

I

~i s : icJ}o

that ~i'

By

~i )

EHMTI~

i

6.2,

P i E J ~[i ~

p

where

( ~ V e S u b u ( 9 ))V@I ~/ #0. the p r o o f Proof

of of

has u n i t : i<~} be

fixed.

i
By

Now

is of u n i t

W~

Hence

~

(][ %" i C P i e J "~i ~ P i e J ~-i ~

6 ~ G w s comps 'p

and

completes

(i)o (ii):

Let

W : u { ~ S (Pi) be

~

such Then H>@,

that

~>~o : ieJ]o

(Vi<j
~ EN~][ e we h a v e

By

(i) we

Let

have

that

Q d @S~W

and

~ "~S (qj)

is of u n i t [S~Rgqil=H

and

J#O

~S (pe)

~J[ ~ ~ let

by and

therefore

Q : o { ~ S (qi)

iil~l~ •

by

where

Let e

[.

[HMTI37.27,

e~=J Let

:

252

7.12.1.

~e

~

•

~i

for

some

~[i ~ ~

Therefore

where ~{E

/J[~

We

Proof

of

Cs

have

and

>~

be

S

zi d ~ 1 0r-l. 1 : xEA>.

Then

and

it

we

let

Then

" iEI}

conclude

NOW

: X.

r l ( l o[ )

Therefore

~UNg(x)

sub-isomorphism. a homomorphism

by

= x

for

every

hEIsm(2J~, ~ ) .

= Vng(x)

proves

that

Cs

strong

and

i 6~

) >~

and

the

g

the

choose

Ti

and

show

by

i
of

Let

that

shows

that

h

that

~Unh(x)

the

>~

Define

"

g Let

xEA.

and

have

: iEI}

of

seen

and

Suppose

~J[

Lct

therefore

definition We

Proof

r. 1 that :

By (~iEI)YiCS

is c o m p r e s s e d .

(ii) :

that

Let

g

hi

:

u U{hix

is /J~ -

Vnhix:O

for

is a s u b - i s o m o r p h i s m . to

a

M

: ~Ung(x)

Cs : x.

if

~i

: i
a sub-isomorphism.

that

a stron@

~<~.

be

: xeA).

Since i
Therefore

Then argument s

h

is

is a

7.12.1.)

of

Theorem

know

that

7.12(iv): the

conclusion

:

is a s u b -

= u{Vnz.x. : iEI} 1 1

showing

is

have

sub-isomorphic

k.l : T 1

is a s u b - i s o m o r p h i s m .

h d < g(x)

h

that

V n z i x i = ~Y n z . xl. 1m : ~ U N z x.1i"

xEA,

proof

notations such

let ~i )

i'

= x. 1

= u { ~ y ! pi) l

above,

we

as b e f o r e ,

use

(~{xEA)Vng(x)=x.

Subu({]L).

every

have

sub-isomorphism.

QED(Lemma

we

is

To

(rllxi)~aYi

that

the

show

we

g d= < O { z i ( x n ~ Y i ( P i ) ) : i E I }

.

that

seen

q i ~ W,

= x

~

then

We

and

VNh(x)

(ii)

Then

can

: VNO{z.x. : iEI} 1 1

: Subu(~

Consider for

h i x i c_ ~S (qi)

a

: icI}

shall

base(~J[)

Y Ik c Id 1 1 -

have

= aS.

T h. e n . z x E C C S b ~ S (pi), 1 1 i --

Vng(x)

We

(ViEI)Yi=U.

Then

by

We

We

Let

V=I ~

show

~•

c HO[ --

r i d: r l ( e Y i ) ~ i s ( ~ ,

Let

to

: i
: i
i

I.

t o be

i ' //i ) "

x, d x n ~ y ( .p i ) . 1 1

= W = u { ~ s (pi)

Then

U

is e n o u g h

(ViEI)VAz.x.=x.. 1 i l

i9

Choose

[HMTI~6.2,

.

ieI.

= ~Y n ~ r -I x9 = ~Y.Nz.x 1 1 i 1 1 1 1

have

: U{x i

Let

g E l s ( O l , ~ ).

Vnzix i = aY.Nz x. 1 1 i we

J=I.

zieIs(~

{ ~

~_>Ibase(~)

C Id. Y i l k l 9 --

that Then

-isomorphism i~I

Then

Cs

By

W o u { ~ s (qi)

and

0~EI

Suppose

~S (qi)

unit

201~ ~

proved

ITi~YiI=~.

unit

is o f

By

(iii):

such

with

~

. c~

of the proof of (i). Y CT,I_1

~i

Assume holds

the

hypotheses.

By

Theorem

if

x ~ 2 I~oBI

which

is

7.12 equal

253

7 .~3.

to

l~USl +

•

by the GCH.

Thus

~=~

QED(Theorem

Remark

The

condition

Let

•

exists,

[IAI~•

~

Let

K

e.g.

~ ~ ~ +~ 3

to any

QED___ t.

If

the

~>~

.

~<6

7.13.2

proposit• such

Proof. Let

~w.

be the

is t r u e

because

assume

Hom(~,~)#O.

Let

~ -doi .

there

qEeZ

y.

Thus

In

cf•

[HMTI33

19,

~

[HMTI33.13,

.

e<•

an

the but

•

I~I

Then

Such

(~/j%e Cs

)

~J~e Cs ~Lf

sub-

or e x t - i s o m o r p h i c

is n e e d e d for

with

K _c I •

that

is not

cardinality

~

to any

such

~AIAI+Iel +

see

the

shows.

Then

since

and

affect

is a c a r d i n a l

: IAI~•

is such

~e.

in 7.12(i).

the n e c e s s i t y

For every cardinal 8>e

and

(V ~ C C s

Let

U = ~I~I +

with

C - d ~Oi --

~ -doi

[Ibase(~) Let

such

is

of

h

-

.

~

are p r e s e r v e d

therefore

and

: ~

Then

IB[=I~.

A : Sg ( ~ ) { x } .

base

Z

~JLEBCs e

: qo ~ q ~ ( ~ l )

with

of b a s e

is

or

Cs

(~qEy)qo~q~(e~l).

and

I>~

by d e f i n i t i o n ,

these

q~(~l)=Z.

there

X ~ {qE~U that

be a

there

Cs

that

C ~ y # 1~

U

~

since

be a

such

base

Let

Then

nH~)

are o b v i o u s

Then

~

(~iE~l)y

q~c~

by

are done.

Prop.7.13.1

cardinal

IAISI~I

Let

]UI>~.

(~i6~1)h(X)

omorphisms.

does

or e x t - i s o m o r p h i c

be any

~IAI

Cs

(~iE~l)X

is

~oB=8

below.

IAI~ '

and

there

the c o n d i t i o n

of

I
~

Then

and we

following

by 7.12(i)

7.13.2.

that

~

that

then

necessity

see

I~uBI.

7.12)

e~

sub-

Let

such

For

: ~Cs

K = {~)~E~Cs

a~e.

~J[el Cs

Cs

as the

is not

is c o n s t r u c t e d

2.

~ S

sub-isomorphic"

To e v e r y

property. of

or

7.12,

7.13.1.

some m e m b e r

a

~Cs~

of T h e o r e m

"ext-

Theorem

Proposition

Proof.

Then

(Discussion

conditio~in

following

~.

therefore

7.12.)

7.~3.

{.

by

Assume

and

let

Assume

= 1~

and

Then

co~h(X)=l ~

under

hom-

yeN

be

O
(~aEZ)q~{y ~[ # ~ .

Then

CoX

W#O

}

This

such

that

Then

and h e n c e proves

that

254

7 .']_4.

IWl>e~ QED(Proposition

3o If

7.13.2.)

HhB

HA!Ai+[~I +

then we do not k n o w how m u c h

is needed.

Theorem

7.14

[HMTI]7.26. properties

See R e m a r k

below

and P r o b l e m s

is a g e n e r a l i z a t i o n

For e x a m D l e preserved

7.15

the c a r d i n a l i t y

under

increasing

the c a r d i n a l i t y 7.16.

of < H M T I ] 7 . 2 5

conditions

the bases

condition

and of part of

are i m p r o v e d

(or subbases)

and

are inves-

tigated.

Theorem

7.14

U d base(~J~). 1)

Assume Crs

Let

~,B

Then

(&)-(4)

O<JUI_<~. ~

is s t r o n g l y

~:Ibase(~)

If

~[

(ii)

If

H_>e then for e v e r y

Let

If

is r e g u l a r

~EK

or

~

•

and a s s u m e

~J~eBK~.

~E

Assume

~>2 I~U@I

Let

Ws,

Cs}.

Then

H:26

for some c a r d i n a l

(3) above. 6Kreg~

or

Then BK

K

BK

and

~ ~K

and

(i)-(iii)

to some

b e l o w hold.

Gws n~

Gws reg,

Gws n~

GwsC~

Gws,

.

.

Let

to some

Assume

1

sub-isomorphic

K e { G w s wd,

6Gws

morphic

Gs,

~Gws

then so is

then

~ _ > ~ then

~>_2I~UBI

Gws,Ws}

4)

~1

(i)

(iii)

3)

Then

if

I~>_~. Let

b e l o w hold.

such that

Gs},

2)

be two c a r d i n a l s ,

K C { G w s wd, Then

~

such that K~{GwsWd~

is s t r o n g l y

rl(~U)

sub-iso-

: Subu(~)

Gws n~

Gws c~

6,

H>-B.

Gws reg,

>~

Subu(~[).

Gws reg,

Gws,

_C I HK~"

BK reg ~ _C I K reg is s t r o n g l y

and

and

BK

Let

_c i xK@'

s u b - i s o m o r p h i c to some

K

be as in

moreover ~ K r~e g

every

or

respectively.

Proof.

P r o o f of

p r o o f of

EHMTI]7.25

easy to c h e c k t h a t

(i):

Suppose

proves

~ke.

the p r e s e n t

the c o n d i t i o n

IAI~•

T h e n the c o n s t r u c t i o n statement

(I),

too.

in the

It is

is not used in the q u o t e d

~K

7.4_4 .'1_.

255

construction. Let

rEU,

Let

z~U

~<w.

Then,

by i n d u c t i o n ,

W ~ {z}U(UN{r}),

f ~ (UIId) r

'

that

preserves

regularity.

(I) is p r o v e d

~.

{~Yi(Pi)

have

IYo[<~ ,

and

f ~

hence

sub-isomorphism,

for

~2

of

and

Proof.

the p r e s e n t

K=Ws.

with

~ICl+6.

Let

Then { fy

(I)

ILI=~,

to some

7.12.1(iii) conclusions

1~

.

By

h

(iii),

let

IU1+l=x<w rEY O.

: xEA).

Then if

Let h

~J~

stronger

we z~U

is a s t r o n g is so.

version

of

(2)

and

~EBK ~

U { base(~

Let

and

6~, ).

Then

such that

and e i t h e r ~

is

base(~

) = UUL

).

K = G w s c~

Then the c a r d i n a l i t y

since

base(~)

construction

,~ )

Let

~ c K

exoept

By [ H M T I 3 7 . 2 5 ( i )

IUI=B.

= UUL.

since

ILl =

[HMTI]3.18(ii)

Ibase(~)~UI=•

This p r o v e s

This

follows

by an

Ibase(~)l

= ~aB =

there

~

is

The rest

all

IUI.

E Ws

follows

by an

construction.

7.~4.1) the h y p o t h e s e s

of 7.14(2) 9 Let By 7 . 1 4 . 1

~ E BGws n~ ~

there

are

~E

.

Let

I = Subh(~][).

I~Gws c~

and

such t h a t

(~Y,ZEl)[fy

= rl~Y(aY)~Is(JSy,

g ~
E Gws n ~

: xCA>o

arid

To see

for some

~(~Y)OI)

(YJZ = b a s e ( ~ y ) n b a s e ( ~ z ) = O )

Let

~.

is r e g u l a r

are s a t i s f i e d ,

(VYCI)~[(~Y)~C Gws ~ : YEI>

of

a somewhat

) >~- Subu(~

easy base-isomorphism

Assume

f~0[

Ws}.

the h y p o t h e s e s .

rl(~U)6Is(~

QED(Lemma

and

K E { G w s c~

easy base-isomorphism Let

for f i n i t e

h ~ ( x u ~ ( x N ~ Y ~ pO))

Let

: Subu(~

Assume

conditio~of

sub-isomorphism

: Po(i):r}I~w

F i r s t we p r o v e

sub-isomorphic

rl(~U)

h ~ < xU~x

Ws}.

or

strongly

is a s t r o n g

be a p a r t i t i o n

I {iE~ Let

(2):

7.14.1.

I~U~l

h

f~eGws

K e { G w s c~

Lemma

: i
(YoIZd)~o

Proof

Let

x=IUl+1.

Z"

T h e n it can be p r o v e d

Let

we may assume

Clearly

: YEI} g

: xEA).

is a s t r o n g

Then

and

and b a s e ( ~ ) C , b a s e ( ~ y ) = Y ~ .

gEIs(~,~)

sub-isomorphism.

for some

~

By 7.14.1 we h a v e

E

256

7. ~ 4.

for all

YEI

that

rl(aY)

(II)

: Subu(~y)

Sb(l~Y) Irl(~y) This implies If

has been proved the

cases

To this end, Since

2j[

.

by

: Subu(~) (II)

above

K~[Gws c~ of

Gws

let

and

Subu(~).

also

~ CGws wd.

So f a r ,

Gws wd}.

It remains to

and assume the hypotheses we define our index set

case.

Let

By 7 14.1 there are

7.14(2)

Gws r e g .

~CBGws ~

Gws n~

>~

Ws, Gws n~

may be not normal,

from the above E I Ws

then

for

and

c rl(~base(41)).

rl(~base(~))

~ E G w s wd

consider

>~- $ubu (~(~y)4ji),

I ~ Subu(2O[).

~ 61 Ws

and

of 7.14(2).

I

Then

differently (~[V ~j[ : VeI)e

fcp< I S ~ v , ~ v ~ J [ )

: VeI)

such that (VV,WCI)
(III)

and

(l~V)A~base(~):V

[base(~v)nbase(~w):base(V)mbase(W), Let

g ~ (U{fv1(XAV)

9 Gws

and

: VeI}

: xEA).

g-i = rl [ (ebase(~[)).

>-~ S u b u ( ~ )

follows

from

For the case

K=Gws,

7.14(2)

Assume

~

9

Then there is Then

i~V

reg.

(~)

{iE~

Let

xTA, kEg(x),

then by

~

(~n) there are

~

iE~

thus

with

let ~ V

fEeT (p) .

there is

I~Vnl/TW=o])3. for some

: Subu(~)

Subu(~

= {i ~ V

~

E

>~: VeI}.

Let

~

is regular.

Let

T { base(~V)

f61

and

pEV.

Now we observe that

is finite.

and assume f+, k+~l ~

f~g(x).

such that

(V#W =

is proved.

is finite hence

f+Tx~g(x)

Then there is unit of

fEl ~

rI(~base(4J[))

(III) and from

: fi~base(V) ]

#k~} ~ [~N(IUA• of

and

hence

gTIs(~J[,~]

We shall prove that

V~Subu(~)

= ST( p )

Then

and

(lUAx) Ikcf. such that

{ic~

k+Eg(x)Ol ~ =x Assume

therefore

ki=fi~base(~).

V~Subu(fJ~)

If. k*(lUAx)Cbase(~J[)

such that

: f #ft 1

1

or

k

1

an8 by regular{ty k~(lUAx)

~ base(~).

By definition of the k i =f iE T = b a s e ( ~ v )

By

7 .~4.2.

(III)

257

above

61 ~ W

then

k 9 ~V

for some

and

f9

V,WCbase(~]%)

(since

hence

V:W).

and by

Then

(III),

k , f C ~ T (p)

k61 ~ V

some

pEV.

and

fEgx.

Fn(1UAg(x))=O

We

(3) These

have

is a c o r o l l a r y follow

Proof

Lemma

of

Then

F ~ {i6~

: f ~k

in

such

for of

that ~

then

Proof.

set

[HMTI37.23.

: ~

>~

assume

(2)

IRg(VsU) I=l

and

y~X

and

for all

below

and

Cs.

c(~,s

: i@I}

(4)

e (~ , y ) e P i ~ i Y v ( y ) i.

(5)

c(~'Y)i

# ~(~'Y)i

and

of

[HMTI37.23.

let

and

F

be any

s : U >~

: iEI),e)

IU/F

-choice

function

sub-base-isomorphism

conclusions

(i)-(iv)

and

(vi)

hold. Le{K,Kreg}.

Then

if

~][EL

the

notations

(VjEJ) IYjIA2.

Let

of the v

proof

of

: X ~ Ij

be

i.e.

=y . ]EF rm sl

then

r:s.

for all

~<~,

yEX.

u~U.

: iEI),

-choice

(3)-(5)

(3)

Let

We use

: y

: icI),~)

conditions

version

is a s t r o n g

f ~ RePc

that

{iEI

< Pv(y) i(~)

=
Gs

.

[HMTI37.23,

If

(F,
proves

is p r o v e d .

of

V=l ~

(F,
eGws

(vii)

9 L

(i)

be an

~

with

( ~ y e X ) Y A P i E I Y v(y) i#O.

PU

the c a s e s

6 : A >-- IA/F

letting

(O)

r,SCRgV

} is 1

f~c(F)gx=gx

7.14(2)

following

is an

hypotheses.

We m a y

z(~,y)

the

Let

there

that

the

of

for

, U=base(~Jt),

I.

(RePc,6)*~

as in the p r o o f

Let

need

be as in 7.14(3).

Assume

Then

is regular.

except

Then

together K

(2)

~J[ E G w s

some

such

Let

~

shall

RePco6

[HMTI]7.23

(vii)

We

[HMTI]7.12.

some

of

Let

on

that

Agx=Ax.

thus

7.32(iii).

(4):

ultrafilter

c

from

7.14.2.

be as

proved

since

3~

c base(4)[)] 1

finite

fe

i

[V#W ~ b a s e ( ~ s V ) ~ b a s e ( ~ S W )

for

and

k =f E ( [ b a s e ( ~ V ) ~ b a s e ( ~ W ) l

~base(~))

clearly

below

for

for all

function

such

that

hold.

all

iEI

uEU.

if

~ (~,y)gy.

Let

c

: ~•

for all

~<~

--

258

7.4_4 o 2.

It is e a s y to see that We s h o w that of

[HMTI37oI2

(i)-(iv)

h o l d by

by the f o l l o w i n g g(V)n~(e*U) ke~U.

h a v e seen that is a s t r o n g

by

(vi)-(vii) :

choices

EHMTI37.4.

If

(iv) holds~

(iv)

The h y p o t h e s e s

Also,

Assume

q = eok,

implies

(3) i m p l i e s

It is e n o u g h

q : <e(kj)

iCIo

Since

c.

this

that

to p r o v e

: j<~>

qEgV.

(iv)

for some

Then

implies

(3iEI)

qE~V.

g ~ RePco8

We

: ~ >~

sub-base-isomorphism.

P r o o f of various

Then

for all

for

follow.

g d = RePcoS.

qc(~(E*U).

k~V.

hold

(i)-(iJi)

Let

: j<e>

and h e n c e

exists.

(vi)-(vii)

(3) ~ so

Let

(c+q) i : < kj

c

and

argument~

~Vo

(c+q)iEV

such a

of

We h a v e to s h o w that

K.

The c a s e s

The c a s e s

/J~ is r e g u l a r

then

K=Gws

Ke{Gws n~

so is

!

g;~

~j[EK

and

K=Cs

Gws c~ by

~

for

are t a k e n care of

follow

[HMTI3706

~ CK

from 7.10(ii)

since

g

is an

isomorphism. Suppose

~EGs

as in the p r o o f particular, E G w s n~

.

of

using

Then

g~CGs

EHMTI37.23,

from

see the p r o o f s

the n o t a t i o n

implies

follows

Q

gV _c u{ Qr

of

(1) and

(5)-(8)

from the proof of

: r~Rgv}

and

(4) e x a c t l y

there.

In

EH M T I 3 7 . 4 ,

(Vr,sERgv)Er~s

~

~ QrnQs:O~

by t h a t proof. Suppose

~J[cGws wd.

r ~ v(y).

By the above,

g ( v ) n ~ Q r ~ ~Qr(q) . (~<~)v(k):r. Thus

Let

kCgV

Let

Let

yeX.

and by

that

have

we h a v e that

H ~ {~
The case since

Ws

QED(Lemma

H ~ {•

: GwsWdNGws c~

and by

for some

ieIo

and

to p r o v e

(1) we have Then

L~F.

Then by

~)l~

rl

: ( c + k ) ~ ~ Pri ~} k e ~ Q r (q)

is finite.

By

(5) we

as desired.

f r o m the c a s e s

K=Gws wd

and

K = G w s c~

.

7.14.2.)

We turn to the p r o o f Let

Thus

follows

it is e n o u g h k~Qr

(c+k) e v n ~ y

: ]< #q•

K=Ws

By

: x<e>,

: (~jcJ~{ri}) (c+k)iC~Yj}. 1

E G w s wde

q ~ < ~(•

g*~CGws

kEg(v)n~Qr.

L ~ {iEI

implies

Define

23~EBGws ~

and

of 7.14(4).

U:base(0i).

Let

Let F

•

~, ~:lul,

be a

8=IBI,

w-regular

~B~.

ultrafilter

on

7 .'15 ~

259

I ~ ~.

We

shall

as in 7 . ! 4 . 2 . -isomorphic s

apply

Let

to

~ -

we define

~ o

~

We

to this

~

(RePco6):':Ol. show that

d

s = < s : ieI)~

yCISubb(~)

is

7.14.2

such that

Then

~

Let

and

Fo ~

~ Gws

.

By

c

and

is s t r o n g l y Notation:

T~Subb(~).

T = PY/F(~).

Let

By

6

sub-base-

for

any

7.10(vii),

IY. I:B

be

set

there

and r e g u l a r i t y

of

1

F

we have ~

~BL

I T I : B P = 2 ~,

~ Gws .

.

Then

Let

by

K

since be as in

7o24.2(vii)

we

21~2B~2

was

7 14(3)

and

have

assumed. L~{K~

~ ~ L

We h a v e

K reg}

proved

Assume

~

.

QED(Theorem 7 . 1 4 , )

Remark

7.15.

(i)

Let

(Discussion

of T h e o r e m

and

Then

•

Proof. EBGws

(J~

~>O. Gws

)(VxCAt~)

Let

xEAt~

I{ z e A t ~

: ZSCo•

(VzNcix)Ez#O ~ c i z = c i x 3 argument.

and

EHMTI37.2,

which is and

: Z ~ C o X } I:~.

But

follows

(Vx~At~)

from

p r o v e d by t h e

O
Then

(ii)

conclude

c.z:c.x. i i

W e do n o t

know whether

(3)

are

best

needed.

possible if

~,~

B=w

and

K

not

The

conditions

of

(i)

IU]<•

not

isomorphic This

dimensional

is as

of

and

is n e e d e d . )

in

then

~g~ _c I •

the

conditions

satisfy ]Ai>•

then

Cs

and hence

we

in 7.14(2)

They

are

c i Cs by

the

of

EHMT]

not

and,

and the

more

(4),

but

and

for

(3)

conditions

of

2)

either. (i)

are needed~

IU]

to a n y

does

-s a r e

by not

Gws n~

c a n be p r o v e d Cs

conditions

~~

(3)

By

1

cizE(Cl{i}~]%)~{O}

x>_B>_u)

can be replaced

if

~>i.

Since

I~I>•

we have

satisfied

(ii)

if

following i

the c a r d i n a l i t y

do not

~J%EBCs ~

are

.

(Of c o u r s e

since

generally,

some

(iii)

C.X~At~{i}~l

(V~c

O
1

1.10.3(i) ,

EHMTI]7.26.)

I BGws ~ 4 I ~Gws~"

) (qx~At~)

l{zeAt~

7.~4

by

More

E~

or

divide with

This

IUI •

base

EHMTI37o22

simple.

precisely,

the

conditions

divides

then

there

~{3 is a

of c a r d i n a l i t y

using

but

the

fact

that

also proves

that

the

BCs

~,

if

finite

260

7 .'I 6 .

modified

condition

possible

for

Ex+~

(1)(ii)

IUI

or

if

~>i,

divides

because

x]

if

is the b e s t

~<~

then

Gws

:

: Gws n O r m (iv)

Let

x ~

Then

and

6K

whether

_c I xK~"

Problems

7.16.

1.

Let

x~2 I~UBI

2.

Let

3.

Let

(See

Is

be as

x>a

6cs~eg

(or

of

in

3).

infinite

Assume

7.12(iv).)

GCH) i s

the GCH.

We do not

needed

know

here.

cardinals.

_C I x c s r e g ?

x~l~l+U(IAIA21auSl).

Is

then

~

sub-isomorphic

csreg?

How m u c h a r e

the

needed

201e6Cs a

for that

6Cs

cardinality

these

4.

Is

5.

How m u c h a r e of T h m

K

the p r o o f

xA8 be two

(/[EBcs[eg,

clear

Let

condition

the

to s o m e

x>~.

cardinality

to be

x>u

sub-isomorphic

conditions

_c I xCsa

7.14

conditions

are

not

the

and

x~(IAIn2 lau~l)

to a

best

xCS~?

(It is

possible.)

true?

the

cardinality

they

are

conditions

the b e s t

conditions

in

(2)

Let

Is

Ws

in

possible. (3)

Thm 7 . 1 4

But

needed?

needed?

how much E.g.

is

are

In

(1)

the

~Gws a c I Gws

true?

Problem the

7.17.

existence

About Gs

~[

x>~+l,

~.

C I Cs reg

true

without

the A x i o m

of

of u l t r a f i l t e r s ?

Proposition with

7.18

below

characteristic

~/~ ~I xGse"

[HMTI]7.22.

Note

Proposition

7.18.

(i)

HSP

Gs

(ii)

Up'

6Gsa

(iii)

HSUp

Indeed,

also

=

the

Let I

•

this

and and

# i 6Gs ~

if .

such

contrast

GS

Cs e = I Cs

O

we note

that

if

that

~

0%~I

is the

between

1<~<~ Gs

~+iCs

(i)

and

then

moreover constructed

(ii).

e>O. HSP

8:]6]a~.

Gs

=

;

Gs

there

if

e>l.

is a

for any in

261

7.19.

Proof.

The case

a~

a<~.

By the

therefore

HBGsa --c S U p g G s SUp

Gs

Cs

-s a n d

To

= I Gs a .

{~, e g G s

(,~) h o l d s

[HMTI37.21

part of

follows

from

(VxeAt(]L)l{zeAt~l

Clearly, if

and

let

first

in

any cardinal

(iii)

~>O)

(ii),

see

(*)

for

is p r o v e d

the proof

g. from

[HMTI]7.4

, B~w.

and of

By 7 . 1 0 ,

7.15.

Assume

[HMTI]7.15

then

SUp~Gsa : I ~Gsa

and

H Cs

= I Cs

(by s i m p l i c i t y

of

which

implies

SUp

.

Consider

property

Cs

= I Cs

(~) o f

0[

for s o m e

Up

below.

: z_
for a n y

8Gs

but

it is

false

8Gs

BZ~.

QED(Proposition

7.18.)

Geometrical was

representability

discussed

in

representability ;Crs ~CA

.

By

[HMT].

The broadest

(known

to us)

[HMTI]2.14

By t h e p r o p o s i t i o n

Proposition Proof. but

is in

To

save

QED(Propositi0n

this

class

Let

~>0.

Then

uses

[HMTI37.2

applied

version

to

representability

of geometrical

C A -s y i e l d s

is s t r i c t l y

larger

the c l a s s

than

IGws

.

is a v a r i e t y .

HSP(Crs

similarly

is c o m p l e t e l y space we omit

to relational

possible

class

this

construction

[N~.

when

below

7.19._

The proof

the

as o p p o s e d

ACA

) = ICrs

to t h e p r o o f

different.

~CA~. of

[HMTI37.15

An outline

of

the p r o o f

introduced

in 4 . 7 . 1 . 1 .

the p r o o f .

7.19.)

8. R e d u c t s

Throughout, We Let

shall ~8.

4]{, e C r s

.

e

and

use

the

Then

rd

B

denote

functions = d rd(~iid)

ordinals. rb p

and By

rd ~

4.7.1.2

rdPeIs(~

p ~)

for a n y

262

8 . "1.

wd Lemma

8.1o

Gws l~~

Let

i
Gws reg,

~Gws,

and

!eL

Gws~

!

Gws~

: 9

(ii)

{rda{~

: 01 Crs B

(iii)

IK

(iv)

I K@ = Rd@ I l<

Proof~

c K

= $RdPl K 8

The

proof

proved

and

if

iff

that

if in a d d i t i o n

0:S8

~EGwsB

xEAo

is a

Let

{pO}USx above.

F=irdP(x).

-regular Thus

Kr -unit

for

Then

IUSrd

any

proved.

I Crs~

= I iCs

by

QED(Lemma

negative

To

part

of

(i)

let

and

EN]

Ih(zdP(do

c

~L

for

is

the case is a

reg eCrs~ r

~

This

K~ _c $ N ~

PropoS(ii).

(ii).

IK B

(iii)

M n @Rd

implies

Crs xeA~

proves

EHMTI]7.14.

)) [#'L

reg Gws B ,

by the

rdPV

and

that

x

-regularn

is regular~

in

above,

Let

since

(ii),

E~iM'ZIl!8~5-6~

since

by the

proves

Kc{ --c sRdPl K B

is p r o v e d

EHMT:]2.1~

if

is CA~

do =!.

=

This

(iv),

8.1.)

Statement it was

(i).

in the

By Lenmla 1.3.4,

4 7oio2(i)

rd x

be as

K~ -unit.

argument

from

K

ioF

0~ )

{a'(base(V)x{(B~

to check,

is

To p r o v e

EHMTI38.1-3,

and

I'=p -'Axo

Hence

: Hd,~l Crs 5

<~rP

rd~x

of 4,,7oio2o

r d P = ~ I s m ( ~ pO%, ~ r d P l

is a

This

Vo

by

the p r o o f

then

rdP ~~

follows

is i m m e d i a t e

of

it is e a s y

is regular~ (i)

~=(~.

let

and h e n c e

Crs B - u n i t

from

or

S u b u ( r d P i O% ) :

Clearly,

x = ~@(iUSx),

follow

the

then

(~-
then

then

s

K=Crs,

RdPK~ _C I K

proves

in

rdPx

For

a-
K B -unit

Jf

B

K ~ G w s reg

i Ol

hold "

~Crs

hypotheses. if

below

,

~ K@, = SNr~ I KB.

extension

if

(i)-(iv)

K~iGws

~

then

K := Crs

: q6V~Subu((}[.) }o

If

Then

Let

~

~ R g p ) l q } ) (rbpq)

"

Crs}

} _9 Crs reg<~

is an e a s y

there

i-] be o n e - o n e ~

'

{rdP:';~P~

and

Gs

o}

(i)

It was

p : {x >+

proved

(i) in

investigate

of

above

does

not

extend

to

Crs reg

if

~>3

as

[N]Prop~15~ reducts~

(originating

with

we apply

notions

the

8.1

Monk)

the n o t i o n s

a:ce e s p e c i a l l y

~ Rd p

A~

Zd

etc

introduced

in D e f . 8 ~ 2

helpful~

Recall

introduced

for

from CA~

below chap.l

in

EHMT3

that to

-

8.2.

263

Crs@-s

as w e l l .

generally since the

to

arbitrary

it o n l y

class

In p a r t i c u l a r ,

of

uses all

algebras

the

ordinals.

Definitions

can

in

found

Definition K

=

-s,

< K

:

for

all

(i)

(ii)

The

By

@COrd)

is

K

: HSP

K

is

K

: HSP

such

said

to b e

RdPK

of

equations

[A]

in

EAN3~.

We

use

similarly all but

for

all

is n o •

note

strongly

8.3.

(i)

KC{i Gs,

Let

strongly

Proof. by

CA

in

to

than

difficulty,

denotes

first the

introduced

ones

below

is

such,

by

algebras

of

of

p

to

CA

[AN2]

: ~ >~

p

for

I Cs

: < i Cs

by

EHMT~2.6.14(i)

iff

systems

one~

defined

@.

: ~ >~

above

definable

iff

equations

one-one

were

understand

similar

equations

one-one

the

we

classes~

: @6Ord),

CA

This

~.

definable

by

is

in

proved

: ~6Ord>

and

simultaneously

a scheme

and

of

for

equations

[HMT~2.6.15.

l<~
i Gs~

NCA

: eEOrd>

scheme

of

equations~

7.18(ii),

of

and

and

that

definable

1<~<~o

Ord

was

and more

any

(of a l g e b r a s )

a scheme

~2

which

Let

algebra~

of

a scheme

K@

< ICrs

Let

by

: ( CA

Bo@-s

without

general

and

equivalent

classes

Proposition

(ii)

are

to

definition

a class

b~

[AN9]

the

classes

c~

any

CAs-s

of

a system

any

in

to

more

of

d_!cfinable

applied

ENJ.

is

be

for

8

notation

We

K

definable

the

~6Ord.

in

K

K

given

schemes

and

for

strongl~ RdPK

results

that

Let

definitions

and

and

type

be

following

a system

~EOrd.

K

The

EAN9],[AN2]

8.2o

similar

similarity

in M o n k E M l ] . be

R~J@c~ c a n

Let

EHMTI]7.16o

I Gs, by

i Crs].

schemes is

1<~. Then

of

a system

Then

K

is

a system

of

classes

definable

by

a

equations. of

I Gs

! Gs

I Gs,

i }Gsl

classes

and i o Gs

i Gs are

are

systems

varieties of

classes

-

264

8.4.

definable for the ICrs

by schemes Crs

and

case

CA

8.4.

(i)

Let

is in

are

QED(Proposition

Lemma

K

(ii)

If

Proof. the

be a system

K,L

by

$(c : d l )l] l

Fact(e):

~ Sb

~

{( F ' , A ' > E I let

CA

~i Let

Z {~(~)

Y

.

Then

~

the

below

by schemes

since

of

equations.

indices

by a s c h e m e

for e v e r y

~w

and one-

by

of e q u a t i o n s

schemes

denotes from

Let

the

y

9

formula

according If

be a formula

E

obtained to

~.

in

from

E.g.

is a set of

formulas

Rd
iff

~A~ F

and

be

an

ACA' }EF.

~

algebra

similar

~

on

i { < F,A>6

such

CAB-s ~

p : ~ >~

ultrafilter

similar

to

Then

let

Let

be o n e - o n e

d= P i E I ~ i IF"

definable

c6id6i~j=l.

: Hi >-- ~ any

is d e f i n a b l e

: ~6E}.

Let

: FCF',

K

be o n e - o n e .

$(~)

equation

Let

B.

be an a l g e b r a

8.4(ii)

definable

Then

of c l a s s e s

[ E YB

6($)

i.

be

from

of the p r o o f

: ~EOrd>.

Let

in

KB ~

x Sb

of classes

An outline

K~ = Uf R d P K B = H S P K ~

( K eL

of

Construction

follows

of c l a s s e s .

systems

is the

<(E)

(ii)

8.1(iii).

B.

Notation:

replacing

then

iff

are

so is

language

by

8.3.)

p : ~ >-

then

EN].

systems

of e q u a t i o n s -one

of e q u a t i o n s

that to

such

B

be o n e - o n e . I

such

I. Let

p"~r1~ c p CA - s .

that

I _9 ~ P ~ ,

For

~Hi

that

Hi ~

d I =

Let

(V< F , A ) C I )

AUp*F and

-2

every

iEI

~ i = ~i~.

by t h e proofs of

let

d~ i

Let EHMT]O.3.71,

O.5.15.

Claim (i)

(ii)

i Let

K

be a s y s t e m

If

~ ~K

~

is e l e m e n t a r i l y

of

classes

definable

then equivalent

to

~P~.

by a s c h e m e

of e q u a t i o n s .

8.4.

265

Proof.

(i) Let

occurring

be

K5 ~

in

e.

Let

a permutation

RdnK8

b

e

Rd K 5 k

be any

6

such

R d n K 6 ~ KB,

n(e)

by

n*H _c e,

i.e.

equation.

i = < F,A)EI

of

by

~i(e),

e

~s

b

be

Let

such

H

that

that

n D ~i.

hence

KS k

n(e)

hence

K~

n(e)

e

be the HCs

set of Let

indices

q : S >~B

Then

>

by F a c t ( * ) .

by F a c t ( * ) . = {i(e).

Then

~

Then

Thus b

~J[

e.

Now

i

{( F , & > e I

: H~s

Proof in the ~.

of

iff

~

p(~)

~P~

~

QED ( C l a i m

of

tion

Up

Let

K,L

~>_w

and

K~

Then

C_ HSP

: Uf

~

proof

let

set of

indices

HCF.

Then

s

Thus

~

~

k

9

occurring

~

in

iff

iff

o~ P

p(~)

: HCF}~F.

and

Rd0KB of

p : ~ >~

~

by C l a i m RdP(KBALB). 8.4.)

HSP

8.4(i). all

be a s y s t e m

and o n e - o n e

of

p : ~ >~

classes. 5

then

is d e f i n a b l e

~->w

p : ~ >~

5

be o n e - o n e .

and

let

Clearly, ~

theorem

Uf

RdPK~

constructed Up ~ P ~

(see

We h a v e

_C K NL

constructed

from

(7[

i and

I c_ ~ 0 ~ .

by

.

Lemma

to

show

Let

Thus

i and

~

Let

by the 2~

8.4(i)

E is p r o v e d .

of e q u a t i o n s . K AL

: HSP

fjleK~NL .

in C o n s t r u c t i o n

We

R d P K B ~ K~.

Thus

schemes

by

f)[ in C o n s t r u c -

by C l a i m

EHMT]O.3.79).

definable

RdP(K6NLs)

_C HSP from

EHMT]O.5.13(viii).

be o n e - o n e .

4][ ~

K

K

classes ~

~_>w

6~ e Uf

by

Let

Suppose

algebra

ultrapower

Clearly,

QEm(Lemma

formula

of e q u a t i o n s .

R d P K S.

the

be s y s t e m s

of

for

Let

EK E

R d P K B : Uf

the a l g e b r a CK~AL B

order

be the

P(~)"

{( F,A>~I

to the

Consider

(KsNLs).

H

P

by a s c h e m e

Keisler-Shelah E Uf

by

first

that

~i

RdPK 5 : HSPK~

show

i.

Let

be such

of e q u a t i o n s .

~][EK .

be any

CA -s.

~

is d e f i n a b l e

to

~

iff

turn

K~ : Uf

a scheme

the proof.

i)

N o w we

have

Let

i = < F,s

~i~i

K

(ii):

language

Let

If

finishes

Let Rd p

Consider

i. T h e n

E IsRdP(KsAL6)

~

c_

266

8.5.

Corollary Then

8.5 9

(i)-(iii)

Let below

(i)

K~ : Uf

(ii)

Mn ~Ke _c Uf

(iii)

~.

Proof.

Let

: Rd~UpKB

: Up

implies

(ii) .

(ii)

Rd K F :

Rd K B

8.5

and

<~
If

as the

following

will

~

hence

Up

not.

Uf

(Cf

(or

and

B_>~I

argument

c O%

By

Rd~GsB

I Gs,

I Cs,

CA}.

and

Then

Rd K B =

EHMT]O.5.13(viii).

EHMTI]7.21.

Clearly,

(i)

Now implies

in P r o p . 8 . 1 0 ( 2 ) .

that then

shows.

of

R d ~ I G s B ~ Uf Rd

I Gs B ~ Uf

Let

~][ w i t h

isomorphic

has

IBI=~

that

B

J UI

there

EHMTI]7.8.

and ~

Rd K

members B

EHMT31.3.15 Then

~JIECA

for all

Thus

but

whenever

.

6Uf Up{ 6{} c

~

is n o n d i s c r e t e

Rd Gs

m_<

is an e l e m e n t a r y

theorem

by

~

by

~<~

UpK : Up

nondiscrete

Rd I Gs

I Gs

Rd

~

for all

be n o n d i s c r e t e .

since

ultrapowers

Rd Gs@.

countable

Rd

theorem

= Rd~IGs B

R d @ I G s B : Uf

Rd~IGs6

#J%CRdoGs B

[HMT]O. 5.13 (viii) ,

proved

by

K,

e ~

_c ~.

Rd~IGs B

does

B>~>O

and

B>~>_~) .

corresponding

[HMT]2.11

Theorem

and

be p r o v e d

Rd Gs B : R d ~ U p G s B

We h a v e

The

7.18(iii)

8.3

we h a v e

Rd Gs 8 : Uf

B>~ I

in the h y p o t h e s e s .

By the L ~ w e n h e i m - S k o l e m - T a r s k i

Rd Gs

Hence

be as

8.4(i),

By the K e i s l e r - S h e l a h

eUfUp

K

by 8.3(i),

89

O<~<~

c UfUp

K6{I Gs,

8.5)

By

subalgebra

Let

R d KF.

Uf Up

and

(iii)

QED(qo
l
R d K@.

(i) by

~

and

are e q u i v a l e n t .

@,B,H

(iii)

]AI>~ 1.

i<@<6

8.6 9

question

) Thm

Let

for Nr

8.6 c o n c e r n s

and

Uf Nr

the d i f f e r e n c e

seems

to be m u c h

between

Nr

and

harder. UfNr

B>2.

(i)

Uf Nr2 ( c s [ e g ~ L f s )

(ii)

Nr2 K B r UfNr2 K 8

~ Nr 2 C A B.

for

KE{IWs, I

GS,

I Cs r e g , CA}.

iGwsCOmp r e g ,

I Cs,

ICs,

8,6,

"]_.

267

Proof.

R

rational

numbers.

p(u,r)

denotes

d= {sE2U

p _d {p(u,r)

Claim

Let

~

Let

Q

denotes

uE22

and

the set of

rER.

Then

~ d ~[2U,

d ~(~)

~c

. Let

(P--{p((O,l>,r)

INr2 Css,

: rER~Q}).

~ @ N r 2 CA B

and

~

is an e l e m e n t a r y

~>2.

O[ ~ INr2 ~Cs B.

Let

SeSb

~

that

8.6.1

Proof.

and r~R).

and

of

and

< S o ( ! ) , S l ( 1 ) ):u and S o ( O ) : s l ( O ) + r }

:

We shall p r o v e submodel

U ~ (Rx{O})U(Rx{I}).

: u~22

d ~ ( J 7 )p

the set of real n u m b e r s

~ ~ < R,+,-,r}re R

B.

Let

uES2

aenote

and let

@

the g r o u p

of reals w i t h

be any f o r m u l a

constants.

in the l a n g u a g e

of

w

such that

the v a r i a b l e s

occurring

in

@

are a m o n g

{x i : icS}.

Then

we d e f i n e

E(u,~)

~ {seSu

Clearlyr

P(S)

: u:pj!oS , R

E(ur@)

C SU

and

In the

~ ~

SU

and

~EpjooS]}.

E(u,~)

~ { E ( u , x i : x j + r ) : ucS2,

s

p

0~(S) {

r~R, ~(

i<j

notations,

~

:

and

if

R

b

(@ ~

%).

i,jES).

~(S))P(S)o

sense of D e f . 6 . O we h a v e

previous

= E(u,@)

~[(S), ~ ( S ) E C s s.

~f~_(2), P:P(2)

and

Note

~Jl :

thatt

01(2).

by our

Some m o r e

definitions:

F(S)

~ {x :x +r, x #x +r z ] z ]

F(S)*

~ {

AH

: H C

: i,jES,

F(S)},

rCR],

and

--W

F(S) :~{: ~ {

VH

: H -C- W F(S) ~}

G(S)

~ {E(u,~)

: ueS2,

G(S)*

{ {

: H C

G(S) }.

Lemma

8o6.1.1.

A(S)

: G(S)*

EH

~F(S)**}

,

for e v e r y

SeSb

B

if

2~ISI.

w

Proof.

Let

ScB,

2~ISI<~.

F i r s t we p r o v e

A(S)

c G(S)*.

By

P(S)

c

8.6.1.i.I.

268

G(S)*

and

A(S)

= SgP(S),

We shall need the following Lemma

it is enough

to show that

G(S)*eSu ~(S).

lemma.

8.6.1.1.1.

(i)

(V~,~EF(S)**)(3~,~EF(S)**)[R

(ii)

(V~EF(S)**)(~EB)(~6F(S~{~})**)

The proof of

(ii) is a simple

Since the proof of 8.6.1.1.1

~ ((~A%) ~ R

~

~) and R ~ ( [ ~ )

(~x ~ ~

elimination

~)3.

~).

of quantifiers

is a straightforward

~

argument.

computation,

we omit

it. Now we show that

G(S)*

is closed under

TRUE stands

for an arbitrary

i.

is closed

G(S)*

(Vgl,g2EG[S))gl.g2EG(S)

universally

*

Let

ul,u26S2

= {O E(uI,~IA~2)

E(uI,~IA92)EG(S) 2.

G(S)*

by Lemma is closed

EG(s))SuNgeG(S) **. TRUE)

:

3. :

uES2, 4.

of

~(S).

formula.

It is enough

and

to show

91,92EF(S) **.

u1#u 2

otherwise

8.6.1.i.1(i).

under negation: uES2,

~eF(S)**.

It is enough Then

to show

SUNE(u,~)

(Vg e

= ~{E(u',

u'cS2~{u}} + E(u,1~) E G(S) ~'~ by ISL<~ and by 8.6.1.1.1 (i). [Su3 ** i,jeS. D~ SU3 +O) : D.. EG(S) for i,jES : Let lj : ~{E(u'xi=x3 • ui=uj}EG(S)* by IS1<w. GCS)**

enough

Let

valid

under multiplication:

if E(ul'~l)'E(u2'~2)

the operations

to show

is closed

under cylindrifications:

(VgEG(S))c~SU3gCG(S)

**.

Let

uES2

Let

jES.

and

It is

~EF(S) *';~.

c[SU]E(u,~) = Z{E(u',Hxj~) J: u'cc[S2]{u}}. Since ISl<w we have 3 S 3 IC i~ 23{u}I<w, and therefore by 8.6.1.i.i(ii) we have c[SU3E(u,~)ej eG(S)** By these A(S)

statements

C G(S)*.

Next we prove

G(S) c_ A(S)

since

= ~{E(u,~i)

: i
enough

we have seen that

to show

G(S)* C A(S).

A(S)eSu /[(S). and

G(S)*ESu ~(S).

E(u,A{~i

Let : i
(~{~EF(S))E(u,~)EA(S).

It is enough

uES2.

By

Therefore to show

E(u,V{gi

= n{E(u,~ i)

: i
: i
=

8.6 .I. 2.

Case

289

i

~

is

E(u,~)eP(S). and

r#O

If

then

= E(u,TRUE). equalities ECm{U}} TRUE)

xi=xj+r i>j

k#i.

If

we have

therefore

: u'CCi{u}-CkiU}}

and

rER.

If

i<j

then

= E ( u , x j = x i + ( - r ) ) ~ P ( S ). i=j

Such a

before,

me{i,k},

i,jeS

E(u,~)

= OeA(S).

kCS,

obtained

for

then

E(u,9) Let

for some

k

and

r=O

exists

by

then

If

i=j

E(u,9)

=

ISIS2.

Using

CmE(U,Xi=Xk ) = ~{E(u',TRUE)

CiE(u,xi=xk).CkE(U,Xi=Xk

= E(u,TRUE).

This

shows

the : u'~

) = ~{E(u',

E(u,TRUE)EA(S),

since

E(u,xi=xk)eP(S). Case

2

~

i<j

by the a r g u m e n t s

we have #xj+r))

is

E(u,xi#xj+r) = E(u,TRUE)

=xj+r)eP(S) We have QED(Lemma

Let

then seen

i(S,H)

the e q u a l i t i e s

By

We may

suppose

obtained

: u'ES2~{u}}

E(u,TRUE)EA(S)

before

+ E(u,xi4

and

E(u,xi=

E(u,xi#xj+r)EA(S).

and let :

S~H.

We d e f i n e

a c SU),

$j[(~) d ~ ( ~ ( ~ ) ) O { i ( S , 8 ) * A ( S )

i.e.

Nrs. ~

Lemma

8.6.1.2.

Then

d {aeD

~S

~

denotes

- (V~6H~S)e

Let

(i)-(3)

(i)

i(S,Z)

= i(H,Z)oi(S,H).

(2)

i(S,H)

: {~(S) >--~ ~ S

(3)

4J[(~) C I Nr~ ~Cs~

: SESb~8}.

the "neat S - r e d u c t "

of

~

,

a=a}.

S,H,ZE(Sb

Then

= {sEZu

rER.

= E(u,TRUE)-(~{E(u',TRUE)

: S1sea}

E CA H.

Proof.

i. U s i n g

and

G ( S ) * c A(S).

=d < {sEHu

~

of C a s e

i,jES

- E(u,xi=xj+r).

B)u{8},

d [@8U,

Let

for some

8.6.1.1.)

S,He(Sb

~(~)

x #x +r • 3

8)u{8}.

Suppose

ISI_>2

and

SCHCZ.

b e l o w hold.

•

It is s h o w n

for e v e r y

= {ss

: SIsEa}

~]~(H)-

= i(S,Z)a.

in [ H M T I 3 8 . 5

~E8~(~2).

: HIsEi(S,H)a} This p r o v e s that

i(S,H)

= {sEZu

: S1(H1s)ea}

=

(i). :

0[(S) >~

~r S Z(H)

is a one-one

270

8.6.<1.2.

homomorphism 8.5.).

(since

By

0~(H) C

to p r o v e

i(S,H)

c A(H). Suppose

v=Su

HeSb

:

and

~(H)

H:~

B.

Then

then

and

i(S,H)E(u,~)

we have

E(SbwB)ui~}, if first :

~

to p r o v e

S1u':u}eG(I{) ~'~, since statements

too,

: {sEHu

It r e m a i n s

Let

: (H~F)IpjlosC_u,

Then

~i(S,H)~'~A(S).

By this,

~Sb0~. A(S).

Then

of

A(B).

: Sg(~

and

C(F )

uEH2

(S))

i(S,H)

and by

--i(S,H)~'~A(S).

We h a v e

By

(i) t h e n we have

: Of(S) > ~

and t h e r e f o r e

(9~H~S) since

I.e.

~eS~S,

c {~(8)a # a

for some

i(H,8)

c(F)E(u,9)6

' ~ S {][(H)

QED (Lenama 8.6.~.2.)

consequence

of

for

_cr,

by

~j~(~) = o r K

by

: SeSb

by

B}.

Let

SE

a ~ A ( 8 ) ~ i ( S , ~ ) ~'~

a~A(~),

there

we may

suppose for some

is

HeSb~

SCH.

i(S,H)='~A(S) = Nr S 0[(H) Then

By

geA(H)~ (since

c 0I(B) i ( H , B ) g

is a o n e - o n e

=

homomorphism.

which

shows

a~Nr s ~J~(8).

(2),

since

~}L(B)~ Cs B.

(2) is p r o v e d . (3) is an i m m e d i a t e

(3%~

Suppose

c ~I(H) g ~ g.

{~(H) g ~ i(H,8)g, = i(H,8) c•

:

Since

is d i r e c t e d and

:

3x(F)~

this p r o v e s

a:i(H,~)g

seen that

Suppose

c ( F ) E ( u , ~ ) : {scHu

S i nce

(i) of 8 . 6 . 1 . 2

already

S,H E

to show

~EF (H) ~'=

: O{i(S,8)~'=A(S)

a~Nrs~JI(B).

aei(H,B):'=A(H).

for every

Nrs ~)I(H) = {c(F)a

and

i(S,8)~'=A(S) D_ Nrs {][(~). then

: u'~H2,

it is e n o u g h

by [ H M T ] O . 5 . 1 1

A(8)

u~=S2

By these

= i(S,H)E(S1u,~x(r)~).

seen

exists

= G(H) ~'=. Let

i(S,H)=':A(S) _D Nr S ~)I(H).

by 8 . 6 . 1 . i . 1 ( i i ) ,

In p a r t i c u l a r ,

We show that

~),

A(S)

c F(H)) .

F = {il,...,in}.

we have

or K

a~i(S,8)~'=A(S)

H~Sb

by

]~S ~)~(H)

K =d {i(S,8)~.= ~0~(S) : S E S b~8}

N O W we show

such t h a t

%)

>~

of

Let

R ~ 3x(F)~[pjooS~}

(Hx(F)~ ~

F(S)

IFI<{o,

By a d d i t i v i t y

where

EF(S) ~'~'~) R b

Next,

~(S)

to show

F { H~S.

9xil...3Xin~

[HMT]O.5.~O.

i(S,H)~'~A(S)C_

: S1s~E(u,~) } : ~{E(u',~)

:

(Va~G (H)) (3g~A (S)) i (S, H) g:c (F) a.

8.6.1.2(i).

to show

and t h e r e f o r e

(and since

i(S,H)

aEG(H) ~'~} by 8.6.1.1.

SCHeSb_ ~8"

[HMTI~ Therefore

by d e f i n i t i o n

i(S,H)='=p(s) c A(H)

IHI<~

seen

S_CH.

HESbm 8"

denotes

it is e n o u g h

i(S,H)~'~A(S) c_ A(H)

S~Sb

of

W

it is e n o u g h ~CF(S).

the c o n d i t i o n s

Nr S 0~(H) = A ( H ) A N r S ~ ( H ) .

U S ~(H)

W

P(S),

satisfy

we h a v e

0~(S) >~

If

W=HS

Thus

8.6.2.

271

By 8.6.1.2(3)

we have

231(2)CINr 2 C S 8 .

Then

{f[= 0[(2)

completes

the

proof. QED(Claim Claim

8.6.1)

8.6.2.

~

Let

uE22.

Proof.

~ Nr 2 CA B. Xu

E(u,TRUE)

~(x)

denotes

O SlClX-

the term

1 9 SoCoX. Lemma

8.6.2.1.

(i)

i {bCB

(ii)

~

(iii)

CA B ~

(iv)

(V~eCAB)

Proof.

Proof

(XIo)

: ~(~

2s(O,1)

of

(i): Then

therefore

(ZOK).

Then

: reQ}

Z = {~{X u : uES} seen

Proof Co(Do1.

~

,

.

since

: b~Xol}

: b~Xlo}i

D2K

Z

since

and

: RIve5

IRI v ~

l{bCK

Recall

rl v

: SC_w 22}U{D01}

by

]rlv*(ZOK) l~e. By

and let by

B~D,

R1 v ~ By

z!m. and

: Sg

rlv*K

By 8.6.1.1

and t h e r e f o r e

Now

is an e n d o m o r p h i s m

Therefore

Irlv~K[~w.

that

: b~Xlo}l>w.

DESu~J[ ! S u ~

~ RI v ~

I~.

VeZ~D.

to show

we have

: rER~Q}.

Z =d {c O~ a, c ~ a : aEA]U{D~I]

Let

to show

it is enough

= {p(,r)

l{beB

D2B

{bEB

~[~s

K { PN{p(< O,l>,r)

clearly

Then

algebra

Thus

We have

Let

it is e n o u g h

the B o o l e a n

: b~X10}l>~.

e Ism(~417;2~,

: bSXoI}I~.

(ZUK).

V d= XO1.

T{bEB

2s(O,1)c2x~T(c2x).

i{beB

~(~(~)

and

= XO1.

)K.

we show

Let

: b~Xol}J<~

of

rlv*

=

we have

Irlv*Z1~w

by

IZlSw.

IRI v ~ l ~ w .

of

(ii):

T

(XIo)

= Co(Do1.

ClXlo ) = C o ( D o I - ( X l o + X I I ) )

CI(Do1-CoXlo)

= CI(Do1. (Xoo+X10))

ClXIo)-CI(Do1.CoX10).

: C o ( D o I - X l l ) : Xo1+X11 . = CI(DoI.Xoo)

= Xoo+Xo1 .

Similarly, (XoI+Xll) "

"(Xoo+Xo1)=Xo1 9 Proof

of

(iv):

is a complete = ~-~2

~

Let

and o n e - o n e

completes

q L C C A B.

Lemma

endomorphism

the proof.

of

O

of

EAN4]

T.]~B~ 2 ~ .

says

that

2s(O,l)

Now

~[ T~2~

=

272

8.6.3.

201 20 P r o o f of (iii) : 2 s ( O , l ) c 2 x <_ 2 s ( O , i ) c l c 2 x = s o S l S 2 C l C 2 X = s o S l C l C 2 x = 20 2 O 0 O = s o S l C 2 C i X = S o C 2 S l C l X = C 2 S l C l X = S l C i C z X , by (iv) and by 1.5.12, 1.5.8 of

[HMT3.

Similarly

(bht not c o m p l e t e l y

similarly),

2s(O,i)c2 x <

<

201 20 1 21 21 2 1 soSlS2CoC2X = SoSlCoS2C2X : SoS2CoC2X : SoSoCoC2X = SoC2SoCo x = 1 1 = C 2 S o C o x = S o C o C 2 X , by (iv) a n d by 1.5.12, 1.5.8 and 1 . 5 . 1 0 ( i i ) of -

[HMT2.

Thus

QED (Lemma

~.

~

E Nr2 CAB.

Let

INolI~W

N u ~ {bEB

and

INIoI>~

and t h e r e f o r e (iii).

Let

Let

automorphism ~V ~.

~

Then

Let

m

{ {p(< 0,1>,r)

: P(O,I)

h : ~

we h a v e

~[V ~

by

= XOI

~ NO2

u@22

Then

XloeNr 2 by 8 . 6 . ~ . 2 ( i i ) -

and

12s(O,l)~NioI>~

show that 8.6.3.1.1 E.g.

D

below

and ~

~V ~ ~

of

P(O,I).

F i r s t we d e f i n e

such that

m~h.

Let

an ~

D = RIvU[

= {xEA

and t h e n by

~ { E ( u , ~ i)

: i
: 9~F(2)~e].

: i
is k n o w n

reR, E(u,

by the

Clearly,

By

= P(O,i).

E(U,Xo#xl+r ) = V-E(U,Xo=Xl+r) , V{~i

: i
= ~ { E ( u , ~ i)

set

P(O,I)

of its atoms.

from BA-theory.

by 12.3 of

At ~

: x~V}.

~

is

is g e n e r a t e d

it is i m m e d i a t e

be any p e r m u t a t i o n

u ~ <0,1).

for e v e r y

= ~{E(u,~i)

of

: rER}.

of

D = {E(u,~)

~

submodel

(P--P(O,1))IfCId].

>~

A = G(2) e

E(U,Xo=Xl+r)cP(O,l )

i
proof.

and

The u n i v e r s e

: i
Lemma

(Xlo)

for

~=

q Nr2 CA~.

>---- P(O,I)

V { E(
E(u,A{gi :

= r~

~

is an e l e m e n t a r y

P(O,1)

8 . 6 . 2 . 2 we h a v e : i
: n~Xu},

X l O = C 2 XlO

2s(O,i)~Nlo

Therefore

(gfEls(~,~))[mCf

Proof.

~[

~ T ~ (Xlo)

such that

8.6.2)

Lemmm 8.6.3.1. Then

~eCA8

: b~X u } = { n E N r 2 ~

then

INoll~w.

8.6.3

Proof.

is a

by 8 . 6 . 2 . 1 ( i ) .

By 8 . 6 . 2 . 1 ( i v )

QED(Claim

Then there

2s(O,1)Xlo

contradicting

Claim

= T(C2X ).

8.6.2.1)

Suppose : ~2

0 1 _< S l C l C 2 X - S o C o C 2 X

2s(O,1)c2x

[S3p.34.

Therefore

:

we o m i t its

273

8 . 6 . 3 . i.I.

~

8.6.3.1.i.

permutation

Let

of

= ~[~V

h

of

~

such

h

of

~[

~V ~

of

h

Boolean since

mCh.

So

We

Let

homomorphism do1.V

= O.

P(O,I)

have

Now we

yeP(O,1).

Then

y~V

y.V

~).

It is e a s y

is o n t o

since

Next we

Then

h

= O

and

since

h

is a B o o l e a n

shall

use

the

as

and

therefore

f m~f,

f(y) f(y)

see t h a t

f

and

= h(y) =

= y.

is o n e - o n e f

homomorphism,

following

m

an a u t o m o r p h i s m

f : A ~ A

is o n t o

and

is an a u t o m o r p h i s m

therefore

to

every

of

feIs(2J[,~/[) and

Then

its a t o m s

there

define

that

f

of

constructed

prove

is o n e - o n e ,

its a t o m s .

to an a u t o m o r p h i s m

shall

fEIs(~,

by

8.6.3.1.1

far w e

mCh._

y~P~P(O,I). that

Then by

that

: yEA>.

Let

generated

by the s e t

P(O,I).

such

(P~P(O,l))If~Id.

BA

can be extended

that

( h(y.V)+y.-V

because

any

is g e n e r a t e d

is a p e r m u t a t i o n

Next we prove

be

At~

~

= m(y).

~

is a

f(dol)

properties

of

= dol V =

= E(u,TRUE).

Fact

i

(Vx<2)(VyEA)EO
Fact

2

(V~<2)(VyeA)

Fact where

2 is e a s y u622

(c y ) - V e

to see.

and

~ c y = c V~. {O,V}.

It s u f f i c e s

~EF(2)**,

and

= E(( O , U l > , H X o ~ ) O E ( < 1 , u l > , 3 X o ~ ) . EF({I})**

such

= E(
c

(y-V)+c

= h(V.c and

(y-V)

y)+-V.c

since

QED(Lemma

h

let

b

IX.

yEA.

= c y

since

y = V.c

is an

~=O,

y#O.

Now by ~

~.

Since

Fact

i for

Clearly

8.6.1.1.1(ii)

But y#O

y=E(u,~)

CoE(U,~) there

clearly

for a n y

we must

have

=

is a

%E

xeF({I}~**

R

~

%.

Hence

I , u I > , T R U E ) = CoVc f(y)

= c

c h(y-V)

y +-V.c~y

automorphismof

(h(y-V)+y-V) = c

since

(y.V)

by Fact

h(V.c~y)

~ V

= c h(y.V)+c

(y-V)

i. f(c y)

=

by Fact

2

= V.c~y

01 .

8.6.3.1.)

Now we prove show

~

3Xo~

CoE(U,~)

and

or

~

R

~<2

•

~

we have

Let

~

that

to p r o v e

that

~

is an e l e m e n t a r y

it by t h e T a r s k i - V a u g h t

criterion,

submodel

s e e e.g.

of

e)i

Prop.19.16

We shall in

[M3.

=

274

8.7.

We h a ve to s h o w that 9(Xo, .... Xn) implies order

and for e v e r y

(3bnEB)

formula.

for some

aeA.

Let this

H

Z --~cpAB >~

6[

Let

P(O,I)nB.

mCf

Since

...,bn_l,a),

complete

of

P(O,I)

P~P(O,l)IfCId.

faEB.

8.6.3

~ Uf UpU[,

we h a v e Therefore

m

f

there

: P(O,i)

since

~J[ ~

c Id

IZl<w,

be an a u t o m o r p h i s m

Such an a u t o m o r p h i s m of

~

and

s

I=

Since Since

is

>~

(P(O,l)nZ)Im

exists,

(Vi
of

exists

9(bo,...

Z1f ~ Id f*H~B

~(bo,...,bn_1,fa )

and by the K e i s l e r - S h e l a h

and t h e n by C l a i m s

By 7 . 1 8 ( i i i ) ,

= Nr2 1 C s ~

= Nr2 Up CS B = UpNr 2 C s B.

E UfNr2=C

for some

E UfNr2~

and

C I Ws~ALf~ follows

QED(Theorem

hull of

Let

~(fb 0 .... ,fbn_l,fa).

NINr2 CA B.

Let

aESgH.

and

and

faEB

8.6.37

By C l a i m

Problem

Let

is an a u t o m o r p h i s m ~

such that

such that

Such a p e r m u t a t i o n

~I

p

9(bo,...,bn_1,a)

the proof.

QED(Claim

(ii)

H C

_c SgZ.

and f

0~ b

be any first

{bo, .... bn_ I} ! B : S ~ ) ( P n B ) ,

P~B~P(O,I).

we h a v e

we h a v e

is

and

{ b o , . . . , b n _ I} ~ SgZ

aESgH

there

formula

3 X n ~ ( b 0 .... ,bn_l,Xn)

~(Xo,...,Xn)

and assume

A = SgP, Since

first o r d e r

Utb

Let

(HAB)U{b O,...,bn_l]

IP(O,I)nBI~

by 8.6.3.1.

and

bo,...,bn_iEB

be a p e r m u t a t i o n

such t h a t

for every

~(bo, .... bn).

Since

~

nE~

bo,...,bn_leB

such that

m*(H~B)

IHl<w,

f][ P

be fixed.

P(O,I)

and

for e v e r y

from

8.20 and by

(i) by

by

B.

we h a v e

[HMTI]7.22 Therefore

~ 6 Cs B.

~[ G C s ~ L f

C I cs[egnLf B

8.6.1-2

ultrapower

Let

q~

IZd~[l=2

EHMTI]6.14

~

by

~ Uf UpNr 2 C s B ~

we have ~

d

t h e o r e m we h a v e

B C K B C CA~.

2<e
does

Cs B =

E Uf Nr2 C s 8.

~(~

)Nr2/[.

IZd~l=2.

and by 3.15(a).

WsBNLf

INr2

Then

Then Then

~[ E

(it is proved.

8.6.)

8.7.

For which

~h~.

By 8.5 we h a v e t h a t

Rd Gs B.

Theorem

8.8 b e l o w

IGs

Nr IGs B : UfNr

Gs

is the first o r d e r

implies

that the

hold?

axiomatizable

first o r d e r

8.8.

275

axiomatizable

hull

a first

formula

Gs

).

order As

Theorem

a contrast

8.8.

Nr~ Cs B

of

Let

in

to T h m

8.8(iii)

see T h e o r e m

1<e<5

(ii)

]< _~ Uf UpNr CA13

(iii)

IGs

~>1. a

The

CA

in

Gs

A x#O.

define

r

some

: A ~ A

Let

8.8.1.

below

are f][

(ii)

For

b

in

8.13.

a~2

language

of

CS8.

CA 2.

0 1 E(SlClX)-(SoCoX)3.

term-function is the

defined

formula

formula

If

by the

VyEy~x

(at(~x)

Let

~

4/

is

term

(y=O

A x~y).

T

V We

formula

~ Z~Yo3

~ Tx~z)

of the

be the

Uf UpNr~

Uf UpCs

~

E V x ( e l e m ( x , y O) ~ ~x~y 1) A

~ ylsz)3).

and

4AECA

.

Then

conditions

(i)

and

(ii)

VYo3Yl%(yo,Yl).

all

proof

expresses

Let

is the

elem(x,y)

Let

XCA

implies

the

with

(?~)*X

c Ate,

existence

of

~(~)(~)*X.

of C l a i m

that

= ~{T(X)

is not v a l i d

equivalent.

(i)

The

is

arbitrary.

to be

at(x)

( V z E V x ( e l e m ( x , y O) ~ x~z) A Vz(Vx(elem(x,YO)

formulas

is d e f i n e d

to be the

Claim

but

5

there

~Ws~ _C K _C CA~.

E H M T 3 p . 4 3.

V y=x)~

be

UfUpNr Gs13 and

=

T(x) ~ ~

see

if

we d e f i n e

term

~[

~_
Gs

(i.e.

~ Uf UpNr CA B.

# UfNr

then

and

Nr

IGs~

is v a l i d

Rd (Wsy)

First

than

which

(i)

Proof.

is s m a l l e r

if

: x~y 0 and

i<~<13.

T(x)

Let

be a r b i t r a r y . CA 7

of C l a i m

y+5

such

: xSy O and

by o b s e r v i n g

that

T(x)

is an atom}

says

that

of

~(~)X

%(yo,Yl )

then

Yl =

is an atom}.

Statement

(ii)

6~ 6 SNr

is i m m e d i a t e

YO = ~ { x

satisfies yke

8.8.1

the e x i s t e n c e

1

8.8.1

of

above.

Statement that

EAN43

~

Therefore

2 of does

EAN43 not

every

N r CA B

N r CA13 ~

states

satisfy

the

(ii)

V yO3Yl%(yO,y I

existence of C l a i m

of

8.8.1.

276

8.9.

The

f irst

~[ e

few lines of the p r o o f of the q u o t e d

Ws y .

a~2

By 8 . 8 . 1 w e c o n c l u d e

the f o r m u l a

~

~

Vyo~Yl~(yO,yl).

(i) . (ii)

follows

be p r o v e d

in T h m 8.20.

c CA

nLf

),

from

moreover

Corollar~

Proof.

Now

(iii)

S

Ws

8.9.

If

~ SPLf

be o m i t t e d

Let

~<w

i<~<~.

Then

csregnLf

QED(Corollary

8.9.)

Proposition

8.10(5)

8.10.

below

Let

xCSl c Rdll xCs 2

(2)

Mn~NUf 8 d CA B = 1 1Cs~

and

Cs nuf Rd C A B =

This proves

Ws

will

c ICs

) ~

~

iff

SPR

CA B = 1Ws +1) _c

Rd a (cs~egnLfB) reg

~ I

iff iff

_~ Nr CA B ~ M r CA B .

l<x<w.

~<m. and

6>e+1.

and

IBlg2 I~1

Cs.

B>e+l. 6>~+1.

I Cs reg.

_c I Cs~

if

~
~m.

By 7.1 and

in [ H M T I ] 8 . 3 .

e<w

and

CA B .

Assume

~w.

HSP~

Nr

C s I ~_ uf R d l C A 2 .

1Cs

~ $ P ( C s ~r egn

[HMT]2.6.74

CA B = 1 1Cs e.

(Cs + l n M n

Rd<~ ~s B

of

Ws

8.9 below.

Therefore

and

iff

csregNRde CA B = I C s Ws nRd

By

cs:eg~Lf

Thus

is q u o t e d

l<e
(1)

Mn nRd

SP(

-subalgebra

N r CA B = M N r CA B .

(5)

since

then

t h e n we are done by 8 . 8 .(ii).

By C o r . 8 . 2 0 ,

Proposition

~e

from Corollary

wWse _c I ~Cs~ _c S U p ~ .

Rd

(ii)

by E H M T I ] 6 . 8 ( 7 ) .

7.13 we h a v e

(4)

from

8.9 b e l o w n o t e t h a t if

cannot

be the g r e a t e s t

(3)

~ UfUpNr CA B.

I Nr~ Gs6 = UpNra Gs6

y=e.

follows

Since

CA -s and t h e r e f o r e

Rd ( w W s )

(i) by c h o o s i n g

~yo3Yl~(yo,Yl).

8.8.)

Corollary

2.6.32,

Thus

of

2 show that

.

QED(Theorem

About

w Ws y ~

is in the l a n g u a g e

Rda( Ws )

c IGs

that

Statement

Let [HMTI]

by T h m 8.8.

c

277

8. ~O. ~.

8d~(cs[eg~LfB) Rd~(Ws

Proof.

Let

Then

~

>

proves

l<~<m.

i<~

generated

by

~

_~ |GwsCOmp~ reg

Vx(x#O

For every

~

~Lf8)

_~ ICs reg.

Proof

~ CoX=l )

~ ~ .

First

CA B b

~x(O<X<do1

A O#1]

~ at(dol) ,

8.10.1.

where

Ax=l

Proof.

Let

0<~<8 where

contradicting

BA

(C7).

RdlCA 6 ~

and

~HRd

~ ~~ .

be the

Cs 2

(Vi<j
(Vi<~)COai=l.

Thus

(Vx(x#O -- CoX=l)

~ 3xatlx))

Cs 1 where

A x#O,

cf. Def.7.3.1.

CA B ~

[Vx(x#O -- CoX=l)

Cs I b

HRd

CA~. Let

Suppose

CA B b

Vx(x#O - CoX:l).

(3X(CoX#X)

A

Hence

: i~--1}.

Then t h e r e

~CA

z d= dos.

~J[

are

Then

aBA

0~

But

and

= h(b-c~z)

~)(h(z))

and

We have i<~,

Coh(Z ) = h ( c 0 d 0

this p r o v e s

B

(~b~B)[b~c~z

is n o n d i s c r e t e .

and for any

non-discrete.

~ 3xAx=l),

CoX#X A A{cix=x

= h(ci(b.c~z))

= i

atoms.

Cs I ~.

formula

~)z

s

~

We p r o v e

cia : c i h ( b - c ~ z )

By

by

B~2.

by

(~-ary)

h(c~z)=l,

~ ~4A.

[Vx(x#O -- CoX:l) A O#1] -- 3 x a t (x).

Then

=d

~

and

Let

Hence

3xat(x)

~

Let

by

Cs 2.

and

generated

~) ) : j<~}

(y=O V y=x))

O<~
~

also

we have

6~)(h(z)).

~

by

by [ H M T ] I . 3 . 1 9 .

for o t h e r w i s e a=hb,

- CoX#l)

is the

hBHo(~,6~) ~b=O]

Vy(y~x

thus

~A e C s I

{a i : i
for any a t o m l e s s

Let

is a

RdlCA B ~

CSl~Uf 8 d l C A B ~

Lemma

~

C s I 5 Rdll

we show

is the f o r m u l a

9 Uf 8 d l C A ~

Let

(~xeC~{O~)CoX=l

seen

at(x)

Therefore

and

Then

Also,

We have

Uf RdlCA B.

(1):

a i :d {< j , j + i ( m o d

let

{a i : i
~i s "

of

~

hz#l,

say w i t h : a,

) = i

shows

06

= i.

QED L e m m a ( 8 . 1 0 . 1 . )

~

By

EHMT32.1.22

N

VX(CoX=X)

Clearly

Similarly, order

iCs

we have

by 8.10.1.

_c M n N R d C A B .

Mn

~

Hence

if

~<~.

IAIN2

We have

M n ~ U f Rd CArl = I 1Cs ~

formula

13xAx=1.

if

seen

~<~,

Let

by

{~BMn

f..~eMna,

Mn~ARdCA

since

NRd

CA B.

thus

4.J[ ~t 1Cs a.

B = l lCs ~ 9

Ax=l

is

Then

a first

278

8. i 0.2.

Lemma

8.10.2.

R~ CA B ~ where

Let

1<e+1<6.

~X(CoX#X)

Ax=O

Then

~ ~ x ( x # O A x#1 A Ax:O)

is the

formula

A{c x:x

: i6a}.

1

Proof.

Let

I<@+i<6.

~X(CoX#X) A

then

Let

~

z d= d ~,~+i ~

: ~ r i

~ by

~ 6CAB.

if

~[

Clearly,

zr

and

for some EHMT]I.3.12.

z=O.

QED (Lemma 8.10.2.)

Let

l
8.10.2

hence iCs

Similarly,

if

and

(3).

using

then if

if

~<~.

Now

U[EDind seen

2Cs

c~
the

xEA

(4) :

Thus

Dind nRd

CA 5 = l lCS ~ 9

since

Ax:O

_c Cs reg~ _c Dind~,

I Cs N M n ~ N U f R d o C A 6 _c I C s ~ N M n ~

_~ I ICs ~

parts

proof

of

of

(2)-(3)

will

is

a

Ws~ _C D i n d

"if-parts"

negative

the

by

. ~ E I 2Csc~.

the

Let

Then

4.7.1.2.

by 8.2(i), Let

rl(~u)*~eCs

This

follow

by

~)LECsa+INMn +i . c(~F)x = C (~F ) X

Then

9

A(f~)c(F)x=O

by EHMTI]5.3.

~

is simple.

# J [~_ ~ s r e+g1

since

U6Subb(~). reg

Then

C(F)XC{O,l}

( ~ r --w c c~) C (~ F ) x E { O , I } . ~ E G s reg~

Let

IA(f~)c , r ~xl_
Hence

9

0<~<6.

F d= anA(6~)(x),

and

EHMT]2.1.22.

Of

, (2)

by 8.5. from

(4),

Then

by

Let

EHMTI]3.16.

Thus

~

~

d ~f)[.

and

A(~) c(p ) x _ c

by

~)[eMna+ 1

We have ~

seen

:d rda* ~ "

[HMTI34,1,

rl(~U)@Is~

Let

and

since

~

~

and

(VxEA) Then ~ ~

by

is s i m p l e

{)[ ~ ICs reg.

We nave

and

seen

( C s a + i n M n +i) c I Cs reg Assume

T6f {TL e ICs by

~<~

and

c I Cs @

~afj[ e ICs a if

I 6 I_<2 I~ I

by 8.1(i)

"f~-E.csregnLf

Thus --

We have

VX(CoX=X)

[HMTI]7.13.

C_ A(~)x~F

Rd

12 a n d

Ol P

Dindc~ntlf RdaCAB = I 1Csc~'

Then

P r o o f of Let

Then

a
~_>~ then

6=c~+1.

Let

EHMT]I.3

CA 5.

CAB.

formula

If

by

nRd

c_ DindanRd

Csc~ c_ D i n d a

and

s

IAI_<2

Clearly,

first o r d e r

Let

~ _ < ~ and

and ~

Let

s ~. c s r e g n L f ~. A D D

EHMTI~7.21.

6{

has

by 4.17(i). [ 6 I_<2 I~ l

Assume

X<~.

a characteristic

We have

If

seen that

Then and Bd

X>_w then IAI_<161_<2 ~a~/l. e Lf

(cs~egNLfB)

C

8. &&.

279

Proof of ~

(5) :

(~)X.

then

~

Let

Then

~ICs

~

by

We have

Since

seen

Let

~

~ ~6

and

~ Dind

~

atoms

~

,

~

is

of

a

x ~ {s~6~

Let

by pEB~

8 9 1 0 ( 4 ) 29

X~it~

~.

Suppose

8.12.

Let

~B.

~.

(Vi<~)a~doi. this

iff either

means

Thus

~.~ ~ l C s a.

Let

~B.

If

then this implies

If

Let

Then

~)x=O Rd

This proves (p))~]~

Then by

~)x=

(cs~eg~LfB)

Then

A(~)y=O

~5 ~ W s 6 n

shows

%~5

~

~=B 0%.

We have seen that

is

161~2

needed,)

: eEOrd>.

21~I=2 IBI

~

2 [~] # 2 IBI

by 8.10(5). Suppose

21~I=2 IBI

w~.

Let

2j[ ~ ~ ExGs~ALf~

Hence K

BArn. Let

~

~ELf

~5 CICs

= SNr Then

Suppose

~<~.

~ ~ ~B2,

x<~.

Then

has

~2 I~I

has non-

by 4.16(i).

K B = SRd K 6

2 ISI > 2 I~l

~ E

by 4.7.1.2

Therefore

and by

, ~<~. Thus

Let

by EHMTI]7.21.

21B1=2 I~1

Rd K 6 C K

Suppose

and we are done.

by 4.7.1.2(ii).

~ ExGs

K B by E}94TI]8.6. Assume

or

: ~EOrd>.

Then

subbases

by

a+8<~

~ EICs

then

by our assumption

zero characteristic

condition

K : < ICs nLf

K = < ICs nLf

{ rd~P0~

~I ( B ~ ) 2 L

R%K B ~ K

~4][.

~(~){x}.

2 I~I = 2 IBI

Proof.

~

~

d ~(6

Then

Let

Ka = SNra K6

E~CsBnLfB,

and

161 S 221~1

(ii)

then

By

~ ~

~0,

~

~

(Obviously,

iff

C SNr

~<~

is the cardinality

K~ = SR~ K B

subunits

If

~

~J[~

reg

(i)

has

~

and

To what extent

8.1~.

and 8.1(i).

Let

such that

x ~ {O,i].

Problem

~<~

o~

: s(~)=O} Let

8.10.)

Proposition

and

characteristic

QED(Proposition

in

in

y { xnBx (p) ~ {O,i}.

D IGws c ~

needed

by 1.4.

~# 8 , X { A t Z

(x=O V c(~)x=l).

~ mind

D ICs reg.

n Lf 6

~

by EHMTI34.1.

Thus

mind

~

~ {

Rd Cs~ eg _~ ICs~.

~[ ~ cs~egnLf~ :(~].

Let

is regular

there are more than o~ ~ I ~ C s ~ "

i<x<~.

by

since BA~.

Z ~ Subum),

If

K

c

B<e

#J[ { ~(s

280

8 .{3.

Then

~J[6CsBnLf 5

e2Gws ~

and

Zd4A

~ Nr

~

will

has

6%.

N r K B _~ K S

if

with

the

~EGws

of e a r d i n a l i t y

is not

nl T ~ {%.

~
We have

~

Then

21Bh>21~t :

compressed.

21~I#21 BI . L e t

Z c

since

seen

E

that

K~ ~ SNr~ K B

Then

8.18(ii).

cardinality

lale[BI

8.13.

Let

(i)

Uf U p C s

(ii)

Uf UpGwsCOmp~

(iii)

Uf U p W s

To p r o v e

and

~

Assume

8.12_)

ZFC t h a t

Theorem

Therefore

in

QED(Proposition

IZl:2 I~I

an a n t i c h a i n

w~

be proved

About

and

condition

but

of

8.12

2 1 a 1 = 2 [BI

note

for

that

some

it is c o n s i s t e n t

~,B

~
iff

r e g = Uf U p R % G w s ~ ~ D_ Rd WS~

Theorem

8.13,

we

reg

iff

shall

~>w. iff

a>_w.

~->~.

need

the

following

definitions

and

lemmas.

Definition

8.13.1.

relational

structure

I

are

four universes,

D

: I•

the

~ A,

+

Convention: = U,

etc.

By a Crs-structure

: A•

Let We

~=

shall

such that

and

ext

C

-

: A ~ A

be t h e a b o v e omit

a four-sorted

< A,U,V,I,ext,E,C,D,+,-,I}

~ A,

~

we understand

the

: V•

~ U,

and

E~V•

~ A,

IEA.

Crs-structure.

superscript

: I•

A,U,V,

Then

if t h e r e

A q~[ = A,

is no d a n g e r

U ~[ =

of

confusion.

Next we define

some

axioms

in the

discourse

language

of

Crs-struc-

tures. Definition

8.13.2.

we use

convention

the

variables

of

Consider

sort axioms

About

the discourse

that

I; x , y

s,z are

(SI)-($9)

denote of s o r t

below.

language variables A

and

for of

b,u,v

Crs-structures sort

V;

a r e of

i,j sort

are U.

8.13.3.

281

(51)

(Vs)(Vz)E(Vi)ext(s,i)=ext(z,i)

($2)

(Vx)(Vy)E(Vs)(E(s,x)

($3)

(Vx)(Vs)(Vi)lE(s,C(i,x))

~ s=z].

- ~ E(s,y))

-

x=y].

~-

(3z)EE(z,x)

A

(Vj)(j#i

~

ext(s,i)=ext(s,j)].

~ ext(s,j)=

=ext(z,j))]]. ($4)

(~s)(Vi)(Vj)EE(s,D(i,j))

(S5)

(Vx)(Vy) ( V s ) l E ( s , x + y )

($6)

(Vx) ( V s ) E E ( s , - x )

(57)

(Vs)E(s,1).

(58)

(Vs)(Vi)(~u)(3z)Eext(z,i)=u

A

($9)

(~x) (Vs)(Vz) (E(Vi) ( C ( i , x ) # x

- ext(s,i):ext(z,i))

=ext(z,i)3 Now of

Crax

(1) -~ and

(2)

I W%

.

Then

shall

str(~)[)

every

d: < A , b a s e ( ~ ) , l

E ~ {< s,x>El~Jt•

t

i,jE~,

sEl ~

(SI)

:

that

Cyl

(str(fA))

Lemma

8.13.4. and onto.

: V W[ >-

and

consists

, ~ , e x t , E , C , D,+

~

: SEX]

xEA.

the

from

Let

eU ~

I

C(i,x)

Clearly,

Bo

the o t h e r s . for

,

C ~iX

str(~l)

cyl~

.

Cy<(~T[) di3

types

of

the

Let I

iW%

Sb~U Tit

:

are o n e -

d : cyl~"Cy~(~ql).

CA -s to be

It is i m m e d i a t e

identical and

by t h e s e

O

~ : a >---- I

with

are

definitions

n=~IId.

Let

r

cy{

: < s , x > E E ~%}

operations

and

{ < A D~ , + ~

d: D ( < i , ~ j ) .

: A T/[ >~

Cyl~(T/i)

f)lECrs

{ ( $ 1 ) , ( $ 2 ) }.

~ ({cy<(s)

~9~ be a C r s - s t r u c t u r e . ~eCrs

T~ ~

and

cyl~ and

similarity

= f][

Let

: xCA>

and

in any

that

We d e f i n e

(52) 9 We d e f i n e

because

by t e r m s

onto.

sEV Ta)

and

consider

CyI(Tf[)

and

such

c i d= < C ( ~ i , x )

cy~

definable

one-one

Cpax

($I)-($7),

CraxU{(59)}.

~ s(i)

be o n e - o n e

: i<~>

by a x i o m s

of

of a x i o m s

Crs-structure

where

{<ext(s,
that

for

Tf[ be a

ci'dij)i,jE~

We

(3i)ext(s,i) =

-structure 9

: ~ >~

-one

A

~-- E(z,x))).

Then

ext(s,i)

~ d~ 13

Let

xEA~>

(Vj) (j#i ~ e x t ( z , j ) = e x t ( s , j ) ) 3 .

to c o n s i s t is

~][ECrs

where

Crs

V E(s,y))].

1E(s,x)].

(E(s,x)

Rgax

(E(s,x)

8.13.3.

D(i,j)

is a

and

Let

,i~>

----

is d e f i n e d

(51)-($8)

Definition

~

~

be

282

8.1 3.4.

(i)

~% ~

Crax

str(0l)

(ii)

b

' ~ ~ Cpax

(iii)

~

implies

~

str(0t)

Proof. -one

Let

W ~ 1~

.

~

~

Then

(i)

write

= W~cyl

(x).

d ij (~)

T/[ ~

: sCV

,

to p r o v e

: seV} with

~

p = cy~(s)

Let

d q = cy~(z).

by

~ : ~ >~

W1 b

I

be one-

{(Sl),(S2)}. cyl[EIs(~

Let

,~).

]~ Let

($7).

EHMT3p.28,

ECrs<.

and

By

iff

Ts

that

such that

and

By D e f . 8 . 1 3 . 3 ,

i n s t e a d of

F

x,yEA

< s,x>{E}

< s , x ) E E ~[

is proved.

: {cy<(s)

cy~(s)

~i

Cpax.

: (ext(s,r ext(z,6i):u qeW

These

9

we

($6)

: {cy~(s) : sex} : cyl~(x+

<W3 , cy~(s) E Dij

iff

EW3 : m13 . c i( ~ )cyl~

prove

~ EGws c~

that

By

($3)

(x)

=

~5 E C r s

seV.

peW By

and

(Viii) (j#6i - ext(z,j)

and

i q = Pu"

We have p r o v e d

and

($8)

uEU.

there

i,jE~.

Then

and

W C--C i[Wl (Dij [W] ~W)

the f o l l o w i n g

Wc~u.

since

(VpEW)p(i]pj)EW

is

= ext(s,j)).

(*).

(VpEW) (VuEU) ( V i E ~ ) p l E w

~"

To this end, we Let

for some

By

(SS) we have

Hence

statements

GwsC~

: i

By

: sEm(~i , [j)}

We p r o v e is a

i,jE~.

: sEV}~{cy6(s)

sEm(~i,~j)

~ cyi~(x)

E ~[w]

and

= cyl~(-(~9%)x)

= {cy~(s)

($4) we h a v e

W=cyl$(1)

Then

Let

= cyl<(-(~)x)

= cEW]cyl~(x) i "

zEV

and let Then

-(~ ) : B]W~

x)

Assume

statement

Craxo

= cyl~(D(~i,~j))

sEC(~i

and

(][ECrs creg.

~ Cyl~(~).

Hence

: cyl~(C(Ei,x)) (2)

Cyl~ ("~)~Crs~ reg

if

reg

reg

Crs-structure

= cyl~(x)Ocyl~(y).

we h a v e

s

- ( ~ )cyl~(x)

: {cy{(s)

(~':)

if

Cyl~(]]l)~GwsCOmp

sex.

: sE-x}

Then

implies

F i r s t we p r o v e

hence

and

~][~Gws c~

In a c c o r d a n c e

($I) we h a v e

have

if

W = {cy<(s)

Convention: sometimes

Rgax

Assume

and

and

Cyl~ ('~)~Gwsc~

implies

2F~ be a

and onto.

d = Cy6(T/()

+y)

Rgax

T~ b Rgax

(iv)

and

Cpax

~ CpaxURgax

str((1)

Cyl<(971)eCrs~

Crax.

str(~[) ~

Let

implies

by

(~).

8.13.5.

283

We show

siWCW3 --

(operations in

i ppj : cy~(s),

say : pi

and

and hence By ( * )

Crs creg

prove

~

CA

W

is a

we h a v e

was defined

ECrs creg.

Let

q=cy~(z)

Let

p,qEi ~

z

so that

then

p :

base(Y)=U.

for

some

s,zEV

Let

ciy#y

j:~i.

Then

EIs(~ , ~). (S9)

Thus

then

(4)

and

1.

ppjEW,

ext(z,i)

cy~(z)CW.

Hence

y,s

and ~

Assume

By

:

Thus

W

is

a com-

~

z,

and

d = base(~) str(f][)

, V d: 1 ~ ~

Crax,

~ q~x.

Clearly,

q~x

we h a v e x

~

($8)

Then

and

yE

p=cy~(s),

C(j,y)ey. since

cyl~E

ext(s,~n)=ext(z,~n). zEy.

Therefore

~ C G w s C O m p reg,

sEy,

by

(2)-(3).

~ eGwsCOmp r e g

str((~),

if

for some

c~i xCx

w h i c h refer to

let

#J[ECrs

It is immediate by D e f i n i t i o n s

str(~)%)

We s h a l l

is cregular.

Then hence

j~I

hence

(2)-(4) we proved those parts of 8.13.4

To prove the parts r e f e r r i n g to

Rgax.

x:cyl~(y)

suppose

= Cy~(T/[),

CpaxURgax.

GwsCreg:Gws reg

Then

~

(Vi~{n)uA(~)x)ext(s,~i):ext(z,~i).

We have seen that

~

~

({n}u~)x)Ip

that

by

AssUme

nE~.

such

zEy).

pex.

By 1 . 6 . 2 ( i ) ,

1.6

ext(s,j)=ext(z,j).

(sEy ~

which implies

in

be such that

(S9) to

if

choose

Then

Gws -unit by [HMTI]2.1(ii)-(iii).

that

xeB

In order to apply

By

(S8) again,

pEs W, pEeU.

Gws -unit.

(3)

By

Let

Clearly

YESubu(W).

pressed

By

(Vjr

(Vi,jC~)s~W=W 3 Let

sEV.

~@ ~U).

~ ~Gws~ ~

and

Cyl~(~). U ~

8.23.2,3(1) and

that

str(fA)

~

($9)

~ ~Crs creg.

Q E D ( L e m m a 8.13.4.)

Definition

8.13.5.

(resp. T m ( X , C A )) in the d i s c o u r s e elements of of

X

Let

language of

Let t~

tively as follows.

Let

x,

CA -s using v a r i a b l e s

denote v a r i a b l e s of sort

Crs-structures.

is

be a set of variables.

Then

Fm(X,CA

denotes the set of first order formulas

lation functions"

t{x

X

t~(ci~)

~ : ~ >~ I and

tr~

xEX, is

on

A

~,6~Tm(X,CA )

C(~i,t~)% , t6dij

(resp. terms) X.

in the d i s c o u r s e

be one-one. Tm(X,CA

from

) and is

We define the and

)

Let language "trans-

Fm(X,CA~)

~,%EFm(X,CA

respec).

D(6i, <j) , t~(T+6)

Then is

284

8 .i3.6.

is

t~T+t~6,

Atr~@,

t~-T

tr~(~)

is

is

Crs-structure

-t~T.

~tr~

and

such that

is meaningful.

Similarly Lemma

(t6T)

8.13.6.

one-one b

and onto.

tr~(~)

Proof.

b

Let

).

Cylc(~)

b

and

~

QED(Lemma

for every

e

since

Let

~

t~

are in the

(T[t,i>ie I

of

Therefore

b

tr6~.

~ : ~ >-- I

Let

cyl~ TT[ b

).

: A 9Tt

~

(t6~) ( < ~ , i ) i e i )

Let

~eTm(X,CA~).

~Fm(X,CA

cyl~

be a

be

~ ~ Cyl~(~7~). and

t~,

tr~Ek]

This implies

>~-- C

is one-one

that

cyl~

iff

~

~

tr~

~

Then

iff

and onto.

~,

let

S

be the class of all

Crs-structures

~T[ for

IIWII=I~I.

Ws}.

in

Hence

Let

9.

Now we turn to the proof of Theorem

Let

Tf[ .

tr~A

8.13.6.)

For every which

and

is

Then

by the definitions

(9]l)k = ~ ( ~ ) (cyl~ok).

CyI6(T/[)

tr~

the term function

eeFm(X,CA

k : X ~ AWt

~Ecyl~ok],

of

3xtr~.

~1~ be a Crs-structure.

it is easy to check, (~)

is

tr~(~A$)

Instead of this we shall write

iff

Let

t~:t<6,

Clearly

<7~,i>i6 1

denotes Let

is

tr~(3x~)

I ~ IWt .

language of the expansion tr{~

try(<=6)

Let

Ax(Cs)

~ ~.

and

~ Ax(Ws)

~eFm(X,CA

Then

F --~c ~.

~][eK B.

Then

~ Cpax

).

~

and by the L ~ w e n h e i m - S k o l e m of

str(f][)

F K~

~ str(6~)

Let

KE{Cs,

Ax(GwsCOmp

Gws c~

reg)

reg

I

~ CpaxURgax.

be the set of all indices occurring ~

~.

We have to show

E SBnMdAx(K )

theorem,

such that

9

and let

Let

Suppose

8.13

there

by 8.13.4.

is an elementary

~]~ eS RMdAx(K)

and

F c IWI .

KB ~ By

~. ~

submodel Let

~ :

Tg

: e >--~ I d

= Cyl~(T;[). [HMTI]7.17,

By

U[eWs B

be one-one Then

~ ~

~ el K~

in case

for

by Lemma 8.13.4

K#Ws.

we have

the same holds

and onto such that

Let

K=Ws.

(Vs,zeV ~ )l{iEI ~

T/g since

by our assumption

K

T~ ~ ~. ~

~,

Then

FI~ C Id. and by

Let

~

(i). in the proof of

~ eGws cOmp

by 8.13.4.

: ext(s,i)#ext(z,i)}l<~.

Then

Thus

Then

i.e.

~ = CyI~(%T[)eWs 9TL ~

tr~

by 8.13.6.

8. ~ 4.

285

Then

~b

d n = 81Id. Then

tr~

since

Then

Cyl

FI~ ~ ~,

(~)

b

= Cyln(str([~))

~

=

We have seen We have to show ~MdAx(K).

by 8.13.6,

K~

b

~

K

b

~.

Let

~'

~' SlBI

~ ~ Cyl~(]]l)

c ICs B

if

and

~

is proved. Let

K=Cs.

and

Vx(c(e)x=x ~

and

=~.

i~

~

~

since

~

~

b

Then

Let

s ~K~

~'

~

since

~<8.

Let

and

. Let [ ~

Cyl

K#Ws. ~

(~)

tr ~.

=

~

Suppose

K B b ~.

~ str(~J~) E S A

submodel of

try9,

~ ~

~.

~ : B >~

and since

~

with u n i v e r s e s

be an e l e m e n t a r y e x t e n s i o n n~ I be such that F1~Id.

if

K#Cs

therefore

~_ comp s ~ws~

and ~

b

tr ~

where

is an e l e m e n t a r y submodel both of Cyl

(~)=01.

K~

~ ~

if

R~ K B ~

the "if-parts" of 8.13 are prove~. Since

~<~<w,

we have

R d Ws B ~ UfUpGws~ Omp.

(x:O V x:l)).

that

tr 9

~][EK .

since

By [HMT]O.3.82,

i<~<~

equals

KB p

Let

~ ~,

Hence

is enough to prove

8.10(5)

Let

Thus

FI~ .

thus

implies

~ESs.

Then

by

try;

be an e l e m e n t a r y

such that

~=~IId,

hence

~

Ut

of c a r d i n a l i t i e s of

T~ is an e l e m e n t a r y submodel of

Then

8 % (WsB~LfB)

Let

UfUpGws c~

GwsC~ ~

.

Thus it

be t~e formula

b ~.

It is proved in

~ ~.

Q E D ( T h e o r e m 8.13.)

By the above proof m e t h o d more results can be obtained. Prop.8.14 b e l o w is in c o n t r a s t with 8.5.

P r o p o s i t i o n 8.14.

Let

l
and

l<~<w.

(i)

Ws

~_ Uf R% Ws8

if

I~1--~

(ii)

xWs

~_ Uf R % CS8

if

21B~I
(iii)

Mn NUf R % Ws B = I i W s

UfRd Ws

Proof. Assume

Let a~.

B

~

if

or

if

(i)-(iii)

I~I=~.

I<~
If

Let

be the i n f i n i t a r y formula

~

then

.

l~l:w.

(~X(CoX#X) ~ 3xAx=l)

~<w

I ~ 1 < 1 ~ 1

hold by Prop.8.10(2). 3X(CoX#X)

- 3xAx:l.

286

8.14

Proof ~ECA

of

(iii):

be

such

Let

that

l~l:w. 0~ ~

~.

First

ultrafilter F on I. Then + is not I~1 - c o m p l e t e , since 7~

Let

is n

H ~ ~SbI

: I ~ ~

such

be s u c h

that

d y = < H { - d j , j + I : j
by

2ni}

: mSj}

= U{H m

_c {ieI

: j~ni}

~

Thus

Mn~Uf

~Uf Rd Ws

6

= I iWs ~

8.14.1.

Then

I 1Gws

je~.

since

which

.

B>~

and

Thus

Let

x d= < d~( i)

{is

ii6I

_< doj

y/F

~ eWs B Thus

the p r o o f

of

: i~I)

and

~)[ is

{i6I

: j~

S - d ~3,j+13.

9

below

I~l=w

: yi~-dj,j+l}

UfRdWs

~Ji ~

8.14.1

~

we h a v e

0~

Then

F

by

since

: xi~doj}

and

for

(Vi<j<~)H~H=O. z 3

y/F ~ B~{O}

for e v e r y

completes

~.

Therefore

formula.

Similarly,

By L e m m a

B k

8.10.1.

Let

Then

(][eMn ~I iGws a.

Let

by we

that B.

either

UfRdWsB~


have

I iGws

~

(iii)

be a r b i t r a r y .

A U f U p R ~ Ws B = 1 1Ws ~.

Assume

Uf UpRd WS B ~

@>O.

Ws B ~

(VX(CoX=X)

we have

I 1Gws AUf UpRd

we h a v e

(I 1Gws

case

~=O

of

URgH=I

~ { - d f3,j+ 1 : j<~}=O.

Let

by

e-ary

x/F,

(Vj<~)Ex/F

IRd Ws B

~

is an

F~RgH:O,

Rd Ws B _c 1 1Gws

Lemma

Proof.

by

Thus

or

is p r o v e d .

Let

~ F

r F.

Therefore ~ s ~{doj : j < w } = O ~

~.

~

Then

UfRdWs

~(~ ~ I O~/F E Rd CA B

(VieI)ieHni.

: i~I).

/Jl ~

~

F~RgH=O,

that

show

Suppose

some

there

we

.{.

~ ~x(x:O

Ws B p

nUf U p R d ~ W s B )

follows

(VX(CoX=X)

-- Vx(x=O

V x:l)) .

Vx(x=O

Since

V x=l) .

: { ~6CA ~ 1GWSo

= {~O,

If

E~I=~

then

IGws

Since

: IA]:2}

from

V x:l)).

~1},

Then k

VX(CoX=X)

Rd Ws@

= I ~1

b

(O#I) .

= 1 1Ws

RdoWS 6 b

(O~1),

The and

1Ws 0 : Ws 0 = { d~ 1}. QED(Lemma

Proof i<• Then

8.14.1.)

of

(i)-(ii) :

Let

O ~
(Vx~A)Ax~I

/3[ ~ Uf Rd CS 6 <

I~I+~ §

if

Suppose

: i<~>,

by

1.3.3

218~I ~

<

V = ~(O)

(i)

is i m m e d i a t e

and

4A ~

and by

EHMT32.1.22.

P~,I+~

and

{ I/j[/F ~ R d C A

We

for

some

(iii).

~(~V){{p} shall

~][ ~ Uf Rd WS B 5

by

: p~V}.

show if

Let

that

J6~l

ultrafilter

< F

on

8 ."1_5.

287

some

I.

Then

Then

there

is

F

is not

y_<e

=O. (Vi<J
6

and

Let

H n

if

I~ I=~

l

if

tel>w

It is k n o w n

: y ~ SbI

: I ~ y

; 2~

I ~I

lel+-complete

that

there

IgAhl<w3

and

(Vice) Ig-l*il<w,

l~l=w,

let

(F i

If sets

of

~,

and

gi

proved

the

For x(g)

: Y >~

: i<2 ~)

i<2 ~

be a d i s j o i n t It is e a s y

claimed

statements.

jEe

d < a(g(ni)) k<~

let : iEI>.

since

: g(m)E{O,k}JqF g#h.

Then

=hni]

: U{H m

a(j)

{icI by

x(g)/F

let union

sznce

(V~:~Rd ~ W s B)

QED(Pro~osition

to c h e c k

and

be p r o v e d

c_ {ici

Ignhl<~.

For

HRd

~

El{x~At/~

For

We

x(g)/F

: {ieI

: (Vj<e)x_<doj}l= p ~

for :

that : gni=

l{xeAt~5

Cs

:

since

we h a v e

~5 ~ I Rd~ if

Define

: U{H m

~ e l Rd e ~Cs ~

/: CRd

~ IRd Ws

gEG.

such

that

and

really

_< dok

be

seen

Then

let

constructions

g,hEG

sub-

i
: g(ni)E{O,k}}

have

every

l~l>w

Let

=

proof.

infinite

for all

and

Therefore

Similarly,

a short

: x(g) i = x ( h ) i }

~ Cl Rd Cs B . ~O.

F i.

Let

{iEI

and

(Vg,hEG)Eg#h

a(j)eAtU[.

~ dok}

by

here

these

: x(g)i

since

and

IFil=e

Then

URgH:I

disjoint

: y >~

c At~

~ F

8.15 Nr~

below

8.15. WS~

shows

since

in C o r o l l a r y

Proposition

almost

that

~ by 8.10.1.

•

B

if

l~el
V p_
8.14.)

Proposition Rd~

IGI=6

with

~

(ViEI)ieHni.

include

gi

d {~}.

Suppose

<

(i)

but we

: (Vj<e)x_<doj}l_<• jel+w+<6

that

x(g)/F

# x(h)/F

2 jB~el

of

such

and FNRgH:O,

that

Then

is of c h a r a c t e r i s t i c I{xeAtf

such

~

that

I(g-l*o)u(g-l*k)l<w.

: gm=hm}

: (Vj<e)x-<doj } I>_6.

such

be p a i r w i s e

Fi"

every

every

GcYe

for e a c h

e=U{F i : i
is

be

since

Let

HNr

a difference

K B = I Nr~ K B

8.20.

w_<~
~ Rd CAB.

and

1<~.

between

the b e h a v i o u r s

for v a r i o u s

classes

K~

will

288

8 .%5 .{.

(ii)

HS~ K 8 # IRd K B

for

Ke{Ws,Cs,Gs,CA}.

We shall need the following Lemma 8.15.1. Then

H

Let

~

Proof.

1<~+1
i<•

and

~ E Ws 6.

~ ~ % CA 8.

Let

~,8,~

and

~

By EHMTI38.1-2

there exists

Prop.8.10(3),

~>i

QED(Lemma

lemmas.

and

be as in the hypotheses. an

0[ E ~ W s ~ H ~ "

Let

Then

~

~

~s

~JZ~R~ CA 8

by

~>~+1.

8.15.1.)

Lemma 8.15.2.

Let

a
In any

R % CA 6

if

Z{doi

: i<~}

exists

: i<~},

e<6.

By EHMT]

and it is an atom then it is zero-dimensional. Proof.

Let

1.2.10,

~ =

~[~a ~,

a~A ( ~ ) x ,

l.lO.5(ii), QED(Lemma

~ ECAB,

hence

we have

Ax=O

A (~)x if

# 6.

~
If

B>~+l ~

2J[cRd I~J, yej

Let

~ d

: i<~}

(Vi<~)x + S doi.

=O =~+i

~q~(~) since and

H{doi

Let

by EHMT32.3.10(i)

= At~

and by EHMT]

assumptions,

and let

~

and

x+~dA,

i<~

Assume

6:~+i.

~(~){y}

and

a ~ Ig(~){z} since

be any Let

~J[ ~ ~ .

and

I ~ Ig(fl){z}.

(VF ~ e)y ~ c(F)z-

xEA

we let x + d x/I. We show that y = + Clearly, y ~ doi for every i<~. Suppose

d

=

: i<~} = O

by

~

y{I

~

Let

~

z ~ (c y)-y,

J = Ig ( ~ ) { z }

iE~F

w~
Let

by EHMTI35.3,

x~c(F)y + c(F)z r~.

(O), y ~ {O},

For every in

K

~Z

~ ~8

and

4A/I.

Let

then we are done by 8.15.1.

Ws

Then

= H{doi

At~

8.15.2.)

: i<6),

Then

By

xEAt~.

Now we turn to the proof of 8.15. cardinal.

x=H{doi

Then

since

B : Sg{y},

in

there

(Vi<~)x/J S doi,

~Y~(~) is

c z : y+z. A ~ Fu{i}. hence

%1%

JS/J.

yEJ

by

F --wE~

~w.

I~J.

J#B.

Thus

Thus

x6J.

such that

This implies Then

and

by

x~c(F)c + x+~c(F)y by

y = dA-c(F)y

x + ~ d A . C ( F ) y +.

By

FCAC~

in

~

we have

By z. zEI

x/J= 8= Then and

By our

8. i 6.

289

+

+

+

dA-C(F)y

= y

,

in

By

yeAtS,

~

Thus

~SdaCA

(i),

using

i.e.

B

reducts

here

is in

theorems

to n e a t JAN5]

neat

about

neat

works

with

rd a .

We

are

see

rs

~6K[

(ii)

Let

~EK

KB-unit. then

= ~{doi

we h a v e

: i
y+6At~Zd~.

is p r o v e d .

(ii)

follows

from

use

Some

the

notions

of the

-s are p r o v e d

definition

there

a representing

shared

rd a .

rd

properties We d e n o t e all

rs a

this

: x

for any

KE{Ws,

Cs,

rsaeIs(T~,

a-regular.

Gs,

rs eIs(97~

which because

of a

it

Gws B

to

In

U.

Gws c~ for

than

functions>.

a n d any

6)

Then

rs rs a

is a set of

rs a

simpler

by

by

of

algebraic

function

in the u n i t

eS8

systems

we use or p r o v e

is m u c h

function

8-sequences

Let

Then

.

are p r e s e r v e d

: q~x)

~ Sb~U

I
be

by

of n e a t

as g e n e r a l

algebras,

not

of

results

of r e g u l a r

eg.

1
and Gs,

clearly

Gws

algebraic

Gws wd,

some

Gwsn~

~ E K [ eg,

~, ~)

for

some

.

Let

Let

even

schemes.

d rs a : < {a]q

B

A general

varieties.

: SbBU

Let

G ws nBo r m - u n l n

of

that m a n y

Let

eK

by

restricting

(i)

Cs,

(i)

y

arbitrary

by

8.16.

8.17.

K6{Ws,

far,

that

of

applications

particular,

Proof.

(coy-y)~I

does not

reducts

reducts

simply

Definition

Lemma

which

neat

shall

by

So

reducts.

reducts

not p r e s e r v e d

works

8.15.2.

or d e f i n a b i l i t y

about

For

by

and

proved

8.15.)

turn

varieties

y~I

We h a v e

[HMTI]7.13.

QED(Proposition

N o w we

x + ~ y +.

rsax

V

be a

Subb(rsaV)=Subb(V

Gws c~

x~V

Let

GwsWd}.

be such

Then

a

)

Then

that

is r e g u l a r

Gwsn~

and rs a V

rsaV.

a

is a

Subu(rsaV)=rsa*Subu(V)"

~
rs V

Let

is a

If

K a -unit

x s

is

if

regular B

be

Let

V

in

is a

V

a-regular.

290

8 . ~-8 .

Let

V ~ 1~

, ~).

,

Clearly,

below holds

(~'~)

Now

~ [~rs V

a-regularity

(Vq~rs V) Eq~rs x

rs x N r s a ( - x ) : O We have

: ds

Let

some

a.

since

V

e r s a c l ~ x. C f e c i x.

Then

~ ) (ii)

seen

of

Then There

and seen

since

and by

Let

rs a e I s m ( ~ ,

x~N.

Then

(~)

by

rs a xOrs

Axca.

rsa(-x):rs

by

~I fbCgb. i i

by

Thus xWO.

implies

and ~_>2.

g~c 9 Let

hence

rs eHom(~[ , I ), is proved.

a-regularity

by

for

by

there

Gwsa-unit

~Hom(~7~ ~ ,

qe

Then

Then

(i) f o l l o w s

from

a~.

is a

and t h e r e ~

rs

)9

G w s [r eg is an

with unit

a-regular

Indeed,

let

V

such that

~ ECrs~

a d:

~

~

d=

rs a V

such that ~+~,

p

d =


is not rseq

:

i<~

q ~ u < 2 : wNi<w+w>, V ~ B 2 ( P ) u B { o , 2 } ( q )

and

~

q~).

~ ~{p~,

8.18.

Let

(i)

IW s

= SNr a I Ws 8"

(ii)

I Cs

= SNr

(iii)

Let

KE{I Cs reg,

Ke _D SNr~ K B

Let

We

rs e I s m ( ] l ,

: ~Ni<m+w>,

(iv)

qC_

q ~ c l/ rs a x.

O
K~

a-

8.17.)

that

Corollary

(-x):

fb' i gbc v

gbeci x

qersacix.

q la ~ r s a x

(ii)

Then

x

and t h e r e f o r e

i

a

rs a d ~13 _-

q~Ers

and t h e n

c ~ rsex _c rSa c~li x. a. T h e n

Clearly,

gEV

f~x

Thus

V~rs x

i,j<~.

bERggNRgf

is

fb~ci x

for all

regularity

QED(Lemma

x

q C g _ for s ome

for some

rs a x#O

We show that

(Vf~V) (qC_f ~ fex) 3.

c,~ rs x = rs c~x..

since

Note

c l9

fl~x a

- -

have

rsex.

We h a v e

6[.

a

rsaEHom(~[~/~[/[).Let

Gws6-unit.

of

d: ~

u>,<Sb~saV) , u>).

(~'~). T h e n

q~CfEx.

is a

-regularity

~

iff

by

seen

qEc

Let

and

rsaeHom(<SbV,

by

:rs V.

a

~

=

SNr~ K

I Cs

T/~(~V)

B>a>l.

iff

is r e g u l a r

Then

(i)-(iv)

by 4.2.

b e l o w hold 9

8
I (GwsC~

Then

and if in a d d i t i o n

~,<~ then

K~.

be as in

(iii).

Then

Ka,N L f~

= SNra (KsALf~).

m

291

8 .~8 .{.

Proof.

Let

KC{IWs,

f r o m 8.17(i). above Thus

Hence

if

~<m

every

then

~doi}.

B

Then

8.18.1.

Is K

SNr

8.19

commutes

are o m i t t e d . Nr~ S K # SNr~ K h o l d by 8.8,

(i)

(i),(iii)

Thus

6~B2.

~ ~ ICs

KE{icsreg,

Let

and

n~,

(iv) hence

Nr~ Cs B _C I Cs~ X ~ {y~N

by

: (ViE~)y~

.

iGwsCOmp

is yes

if

This can be p r o v e d (K ~ D c

below will

Thm.8.19 with

reg}.

Let

B>~.

and

a<~, from

be u s e d

CH M T 3 1 . 1 1 . 9 - 1 2 .

later.

to o m i t

~>~>i,

So the

in the p r o o f

O nly

under which

$

and

K = Nr PK

and

for all

[HMT32.6.33.

8.19. O<~B.

The

following

a.

8 < ~+~.

b.

Nr Up~A C HSP Nr 01. Uf Nrc~ UpU[

conditions

a.-g.

are e q u i v a l e n t .

for all

~[ E W s B ~ D c

_D Up Nrct(A

for all

9._/[E WsB.

: Up Nr~ K

for all

K _c CAB.

for all

4J[E Cs B.

d.

Nr~ U p K

e.

Nrc~ H 6~ _D H Nr~UL

P

two o b s e r v a t i o n s .

W s B _c K _c CA B PNr

see

the o p e r a t o r

the o p e r a t o r s

whenever

[HMT]2.6.29. of

this t h e o r e m

t h e m are the f o l l o w i n g

Nr~ S K C SNr~ K, and by

Concerning

the c o n d i t i o n s

the o t h e r o p e r a t o r s .

EHMTI]8.6

and that for all

) _C SNr ~ K ~ .

concernes

The r e a s o n s

was p r o v e d

Let

c.

~ ~ ~

= SNr~ (KB~LfB) .

for some

by 1 . 3 . 4 ( i i ) .

Let

the a n s w e r

is w h e t h e r

EHMT32.6.33.

Theorem

B=~+n

By the

KB?

_c SNr~ K B.

Theorem

Then

proving

Let

We k n o w that

question

K~Lf~

EHMTI]8.6.

follows

8.18.)

Problem

K ADc

that

K~ _D SNr~ K B

from

We h a v e p r o v e d

B
B~+~.

Then

follows

we have

a-regular

JXI~

QED(Corollary

=

is

reg].

: SNr~ IWs B

= S N r KB.

Let

Assume

iGwsCOmp

EHMTI38.5 K

(ii) :

~EGws

8.17(ii).

~CA B

IWs

and by 4.1 and

P r o o f of

Nr

i csreg,

B>~>I, Kc

292

8 .'19.1.

f.

Nr~ H6~

--C H Nr~ SUL

for all

g.

Nre HK

= H Nre K

for all

(ii)

Let Up

~<6 Nr

and

SUpK

=

(iii) There are Nr

Proof.

K c CAB. Nr

K

Suppose

and

O<~

Case i

Up~

HSK.

such that Uf

Nr

K ~ Nr

HSPK.

~+~8. an

~J[EWsBADc B

and let

and negative).

{qEBZ (~)

: qo
AX = 2OH.

= ~

Let

and

~(6~)

Z

such that

be the set of all integers

H c (~+w)--e

H1q!~}.

Clearly,

applied to D m ~(4.)I.) H ,

X

Let

E Mn~

s

is small in

A ~ Mn(fA)O{xEA

three unary terms

discourse

language of

%(x),

with

IHl:w~I BNHI .

{ ~@Bz(O)

p(x)

and

since and

By

(both Let

X ~

~J[ ~ ~ ( s

4it and hence,

: IAxAHI~}.

by Cor.8.21(iii)

we define

%(x)

C Nr

~ HS,P Nr

0 = ,

positive

~

SK

~3.

Let

Then

H Nr

and

(i) We shall construct Nr

and

K C_ CA 6

Uf K ~ Uf Up Nr

K _c CAB.

Then

SUpK

a_
{It E Cs8reg ~Dc B.

by Lemma

Hm~=O,

1.3.3,

then

~7]~ ( ~ ) E W s B A M n B.

T(x)

below

){x}.

Next

in the

CA 3.

a 0 1 = c'3"E-(c2x'sls2c2 x ) ) t

i + s2c2x].

a

p(x) = c(2)-(dol.c2x). a

T(x) = %(X).p(x).CoClC2X. Claim 8.19.1. R(q,x)

Let

~ (E2U

antireflexive

and

~CGws : q~Ec2x}. DoR(q,x)

Proof.

Let

Suppose

{,}

then

,

i

q 6s2c2x

, xEA, q E V E S u b u ( ~ )

[[,x,U

and _c R.

and hence

Then

qC<(x)

and iff

U=base(V). R(q,x)

Let

is transitive,

= U. q

be as in the hypotheses. Then

ER.

Let

0 i q, d= qbde012 E c2x.sls2c2 x. Thus

q6%(x)

implies

R d R(q,x). If that

qE%(x), R

is

293

8.19.2.

transitive. Let

beU.

The c o n v e r s e is easy, Then

versely,


q~p(x)

implies

implies

is antireflexive.

so

iff

01E~eo1"C2 x qbb

eR

DoR = U

qE%(x)

for some

R

implies u.

qgp(x)

Thus

( ~ b e U ) (~d" ~ ) q b01e d c2x

iff

is transitive. Con-

qep(x)

iff

R

~ q e coClC2X.

iff

Q E D ( C l a i m 8.19.1.)

Lemma 8.19.2.

Proof.

Let

~

~(x)=O.

R(q,x). R

Mn

Let

b

~(x):O.

~6Mn

.

We may assume

Then

T(x)#O

UESubb(~)

is a transitive,

Thus there is

for some

such that

c2xESg(~/J[)G

a x = c(~)d(~•

there are

~Rgbi~w

or

i,jeF~2

then

{qde'Ol qed}ndij Ol Aa =O. then

= O.

Thus

< e,d>ER.

THis is a

antireflexive.

Hence

Let

and and

iEw.

Thus

Case 2

Let

Then

: i,j~F}o{a

< d,e>ER.

IRgbI~

F --w C and

IHi<w

i,jeF.

and

If

i,jE2

ie2,

jeF~2

then

since

(Vx<~n~)

iff

01 qedEg3

then

iff

O1~ c2x" qed R

by

there is

By

(VgeG)EqdecgOl

since

DoR = U.

: x<~nw}

Let

and if

O1~ c2x qde

we have that

such that

H ~ q*(FO3).

O1 qededij

~)[EWssNDc B

y ~ < C(Li)X (VieH) l{je~

Then

~ HSP Nr ~

~.

By

< d,e>ER

is t r a n s i t i v e and

: iEw).

by 8.19.1

Hence

q~ ~

g e n e r a t e d by

Let

: i~Lj}l<~

~(yi)r

T(~)(y/F)#O. ~/F

U

R

r(x)=O.

n o n p r i n c i p a l u l t r a f i l t e r on

= 2UH

By 8.19.1

Suppose

8.19.2.)

Now we turn to our

: new)

G ~ {dij

contradiction

,01, b

and

By EHMT32.1.17

such that iff

c2xeSg(JS{ ~ ) G

By

QED(Lemma

Ol C qde dij

qET(x)

qe~U.

R.

~
d,eEU~H

Let

(Vi<j<e)< bi,bj>~R.

where

for every

by EHMT32.5.25.

r e l a t i o n on

t r a n s i t i v i t y and a n t i r e f l e x i V i t y of such that

xeA.

be such that

antireflexive

bEaU

{)lEGs

~

IX}.

~ ~Oi/F.

Let

Then

~)(y/F)=2

by

H C (a+~)~.

since

~ HSP Mn

R(O,Yi)

~

F

be a

L ~ < Hn(~+n)

because Thus

~)X

y/FeNr

= {< i,j)E2Z

by 8.19.2.

since we have seen that

Let

:

= ~ .

: i<j}.

Therefore

{)[CMn .

O<~<w~8.

We shall show

Nra Up (WsBAMnB) ~_ HSP N r Mn B.

Let

U[EMn 6

be n o n d i s c r e t e

294

8.9_9.3.

and

F

be a n o n p r i n c i p a l

<doi

: i6e}/F.

by

5>0;

~

and

and

Then

_~ HSP Nr~ M n

Nr~ Up~

(2) We prove Case

i

y6B

c(~)y#y

[HMT]2.1.22.

ultrafilter and

: c(@~l)y.

Hence B

~

~

for all

on

e.

~)y :i. But

Let

~ Z

Mn B .

nondiscrete

{ ~/F

c(~)x

We h a v e

~Mn

and

@ ~a~.

N r Mn 8 ~

HSP Nr

~

Let

y

Then

yEN

= c(~I)X seen

by

that

B.

_~ UfNrc~ Up{)[..

(36~Ws~)UpNr 4A

~.

Lemma

8.19.3.

Proof.

Let

Recall

~_<~

the

and

formula

~+~-
at(x)

Then

from

Up ' N r WS~

Def.7.3.1.

_g Uf Nr~ UpWs8.

Let

9

be the

formula

VxVy(A{O<X~doi Claim UfNr

8.!9.3.1. UpWs B ~

Proof. and

: i~w}

Let

some

since and

By

~ ~)

~.CUfNr

UpWsB.

P~%/F

~

~

Let

A(b/F)=O

iEI

be

P ~%/F

such

: j~Ax

l

e Y(hi)

Thus

~

since

at(x)

and

P~Z/F

b

UpfA

for

~.

Then

discourse by

ECK]

such

: jey~

Then

>.

Let

Let

b __d { d h i , h i + 1 : icI}.

{icI

: nEA(bi) } c H n U H n _ 1 ~ F

that

x #O.

Then

O
~

is a n o n d i s c r e t e

73xat(x),

is a f i r s t

-x <x 9

order

hence

4j[ b

formula.

l

and

WsB~

we m a y

y.EF. ]

Then

be such

b/F e N r ~ P ~ / F

for

since

Let

(Vi<jey~a)

: I ~ 7~

1

Ws

~,

CA -s

since

(VjeBNe)

h

~

B

x/F c Nr P % ~ / F

mRgZ = 0

that

of

Hence

Let

~EIws

P~%/F

Thm.4.2.11,

ieI.

] : jeB>.

some

language

is n o n d i s c r e t e .

1

H
~

[i ~

for e v e r y

H d < ZD~Zj+I

by

P~Z/F

in the

ZE ( Y ~ ) F

is

(VieZ~)ieH(hi).

because Let

we h a v e

Let

~

~+-complete

y d < [iEI

there

Then

Suppose

is n o n d i s c r e t e

l

Then

I.

formula

is not

~.

B>_~+~.

Then

on

E

EZ i _D Z3 _C y.33. that

F

Then

be a r b i t r a r y .

and

(3xat(x)

that

y d : ~+~.

~>~

is a u n i v e r s a l

~_>~.

assume

Let

ultrafilter

~

9.

--TA{O
every

nEy~e.

hi ~ A x

by 1

.

Thus

73xat(x)

by

x/F { A t g g / P ~ / F . ~7~ p ~ / F e U p U i

8 . '1 9 . 4 .

295

QED(Claim

Let Let

8.19.3.1.)

~

be any n o n d i s c r e t e

be a n y

F

Then

~ ~

3xat(x)

nondiscrete Hence

nonprincipal

(let

QED (Lemma

Let

~ d ( O : i<~),

Such a

~(~ ~

since

URgH:I

Bu (O),

since

~[

let

~ ~

~

~

~ ~e

~.

~ w'O'~/F.

since

d < ~("- d j ,j+l Y =

and

Let

c Zd~ ,

jBl+-complete.

is n o n d i s c r e t e . ~

(IGI_>IPI

and

(3) We •

We h a v e

Let

~

: j
is : new)IF)

< {q~6

Then

IPI>IAI . IZd'P 1:2,

I Z d s I=2

by

[HMTI]6.13.

I

such that

we have

be such that

and

~6 = Then

(RgH)NF=O,

(Vj
b / F @ { O ~ ,i~5 ]

(V ~ E U p O [ ) EIBI>IAI Up~][ :O

Let

IBI>IAI:IBI.

be such that

~

since

since

IZd~4~

~J[ I>23.

(~Up~)

~<~.

_<~ Nro~ H~

.

cf(•

(For e x a m p l e we may c h o o s e

=

b d {qCB){ : qo:O}.

: q o : U q ~ ( ( ~ + ~ ) N ( ~ + n ) ) } : n~w)

~t ~ % ( ~ ){w,x n : nC~}.

be such that a<~

U p ~ NNr

such that and

: n~)

By

A(~)(b/F):O

by

: qo=n)

e>O.

b~IA

seen t h a t

~@ B

by

H : B ~ SbI

(96%CCsB)HNr O%e

~ ~

x d ( {q~V

c Up ~

on

Moreover,

be a c a r d i n a l

.)

F

Let

IZd~l:2),

show

~

and

Let

Then

~ Nr Up~J~ .

d ~V,

INr O[ Ik~)

and (Vi<j
Thus

&

for some u l t r a f i l t e r

(ViEHj)bi:dj,j+I].

_

and

~ x a t ( x).

: ne~}/F

){x n : nEi~}.

exists

is not

~:~

b

w

~ O.

by 8.19.3. l.

V d

Z d T ~ [/[ = ZdO%

= I4j~IF

+

on

~

O<~
Old

Let

AtCI~

8.~9.3.]

2

F

~[~ ~

x d= ( d ~n

Case

and

with

ultrafilter

by

~ ~ Uf N r UpWs~

Ws B

and

Let d

x : <
4A ~ SP Cs B ! I ~Cs B'

~

=

~s

n

: iew) b using

,

d

w :

: new).

Let

[HMTI]7.21.

a

Claim

8.19.4.

Nr H~t

Proof.

Let

we h a v e

A(~)Xn=l

if

Xn/J#O.

JeIl~

b

EO#I ~ 3 y ( C o Y = O

A Ay=l)].

and and

Therefore

[ =~ ({~IJ), IRI>I. Let new. In a COXn=O. Thus (Xn/J)~R by a>O and A(Xn/J):1

we a r e

done if

Rgx ~ J .

Suppose

Rgx ~ J .

Then

296

8. ~9.5.

A(w/J)~l 9 J

A(Oi)w

for e v e r y

n
since

while ~

and

new.

for

= iu{~+n

(It is e a s i l y

i~n,

we h a v e

: new]

l=Cob:CoXii

CoW=l

checked

(c + n W - W ) ~ C o ( X o + . . . + X n ) c that

c +nWi-W i = 0

~ Co(Xo+...+Xn)i. )

8 CoW=O.

and

and

Thus by

IRI>I

Thus

if

w/J E R.

we have

A ( ~ )(w/J)=l

c~(w/a)=o.

QED(Claim

Let

8.19.4.)

I d Ig~A {x

Claim

8.19.5.

Proof.

: newJ

n

v = d < CoXn

: (3nEw)a~VO+...+Vn]. f

25 d

(VyEB)E~$)y#I

Let

member

and

of

P

V

a

Then

"

( VnE~)v n C Z d ~ ,

also that,

such that

2A / (InNr {A )

CoY#O] 9

: new). Note

~

f =I

s i nce

for all

thus

Cob=l, i~n

Vo+...+v n

and

f ~O

l

i>n. By

Let yEA

y e N r {A there

kC~

and

:

is the

for all

1

and assume

are

I = {aEA

A(y/I)=l

F c

B

and

c~(y/I)=O.

such that

OEF

and

we have

that

Then

y/l~O.

yCSg(~F

~)L)

--W

{w,x i : i
By

: iEFNI])

Since

y/I~O,

ILI~.

Let

Thus

q~Yr"

rEL

such that

EHom(~6~Fr , ~F

).

: fO=t}.

Let Let

then

and

r>m

are

qey r

Such a

by

exists

: (fUp)Eb]

By

iO q t a ~ Y r.

A ( ~ )h(Yr)=l.

Then

Hence

Therefore

for e v e r y

we have aE~

such that

Let

tE~

be

cf(~)>w.

Let

iEH

and

By

Up*H = t ~

pjr E

rnH=O.

: bEC).

yENr

rEL and

: Yi~O}.

is shows that

~ ~+r.

and

Then

g ~ F1q.

L ~ {iE~

pjr(Vi):pjr(Xi)=O

By

By

(COY+

Vo+...+v n

H ~ (~+~)~(~+r). t

Let

Fn(~+w)

and

k
~ ~ ~Fx.

and

mew.

what

Thus t h e r e

d h = < {fEF~

i qtEYr

Thus

FAA(Yr)=I

and

t > a+u+Uq~H. Let

i~A(Yr).

since

)=I.

u ~ qO

r>m+k

pjr(W)=Wr

r}

Fm~ s

and

d i p = (B~F)lqt.

= {fEF~

for some

be such t h a t

is such t h a t

Let

c (y/I)=O

~ Vo+...+Vm_l

YrESg (s

C8 oYr=O#y r

and

the a b o v e n o t e c o n c e r n i n g

EHom(f}~, ~ ) i
A(y/I)=1

[HMTI]8.1,

we have

and

i~e

gEh(Yr)

h ( w r) =

we have

and

hE

that

g~h(Yr).

( v f E F ~ ) E f ~ E h ( y r)

and

By f~

~h(Yr)]. Let u

and

~ ~ a

(~IId) ua am

such that

Then ~(t)=t.

~

is a p e r m u t a t i o n

By

[ H M T I ] 3 .1,

~

of

~

induces

interchanging a base-

8. i 9.

297

-automorphism

~Is(

~ h ( y r) : h(Yr) above,

since

@ , @ ).

by

By

~(t)=t

h(Yr)eSgih(Wr)}.

~(u):a.

we have

This

We h a v e p r o v e d

~h(Wr)

: h(Wr).

is a c o n t r a d i c t i o n

that

(~yENr

Then

by the

~)[A(y/I)#l

V

V c~(y/l)~O]. QED(Claim

8.]9.5.)

By C l a i m s

8.19.4-5

~Se HNr~~ .

by

we have

Recall

(4) We s h o w that

Recall

that

[~HI~w. (4.1)

by

~2

The

c I Cs reg.

HNr

S~

Since

q~

IZd~l=IZd~l>2. for all (4.2)

z~2

that

The

Ws~Dc

.

Let

V ~ 8 (p) ~ ~(~

p r o o f of

and

and

){y}.

HCe~8

with

case:

to a

xCB~

S~

_~ HNr~ S ~

I Z d ~ ]>2. Ws8nDc B of

as well

H]=~

and

~ E WsB~Dc 8 for t h i s

{][ E

f r o m the A=Dm H

and h e n c e

(Vye

reg Nr ~s B C

29,

Nr S ~

C SNr 411

I~2.

A b o v e we have

Let

~

1]~ ~ .

~

~ ~HNr

S~

By u s i n g with

Then

and

and a

as follows. IHI=~

~EH~

and

Recall

He~=O

was o b s e r v e d Let

A (~)y:H. H

B.

the ideas of 4.12 this

A(•

AZV]y=H.

We h a v e p r o v e d

OtEWs B

A=Dm H H~S,

.

Then

~6 cs[eg~Dc

an

C

and h e n c e

) IZ d ~

for some

(J[ and

exist

Then

) = Icsreg~Lf

(4) that

Then

xEA.

EHMT]2.6

In [ H M T I ] 6 . 1 6 ( 2 )

y ~ x~V.

[HMTI]6.16(2)

By

and

be as in 4.12.

Then

It turns out

By C o r . 8 . 1 8 ( i i i )

] Z d ~ [>2. H~

that t h e r e

(VyEA)]~nAyl<w

4J[ELf .

~eNr

such t h a t

and

then

(V ~ E H N r

lines of the p r o o f

peS~

~cs[egeDc B

for some

H(cs~egALf

Thus

Nr~ H ~

~)t can be m o d i f i e d

Let

is

states

Ax=H

GEl csregALf

Thus

are c o n s t r u c t e d

the f i r s t

~

~eH{~.

that

there

IZd~l>2.

with

By [ H M T I ] 5 . 2 ,

c IcsregeLf

observed

HNr /3[ ~ N r H ~

(3) is proved.

for some

Prop.4.11

HAm:O,

Therefore

.

S~

with

A:Sg{x}

Hence

C Icsreg~Lf

~I ~Css" Thus

Hence

case:

~EH~

of 4.11 t h a t

6 N r A) IAyI<~.

Then

be a c a r d i n a l .

and

[HMT]I.6.4-8.

H~.

6.

~>O.

csregnDc

E cs~egnDcB proof

and

~ ~Nr

~

~ HNr

~Ws6~Dc

~A~+~

Let

that

Nr H ~

as for some

that

there

~

~ ~V

~6H~

etc. there.

and

By r e p e a t i n g exists

from

the with

298

8.19.6.

IZd~I>2.

Now

WsNDc-case

as

~Lf

.

By

for

) :

So

far,

any

(i)

change

Ws~ADe~

the

"only

and (ii)

Proof.

yCE

there

then

C(Ay~)y This

Let

0.2.23-27

h ~

we have

have s

of

HNr ~ = ~

If

A : Dm ~

the

the

proof

~22

that

of

(i).

.

Let

SECo~

By

of

~b,g ~

3.~5(a),

(4.~)

above

H~

Nr

~

~

such

~)/R, and if

hypotheses.

we have

of

g HNr

$~

~

and

]~r

that

R :

SN2N

~a(~/S)), I~I<~

then

Then

Let Hs

c ([.) z e N

IB ~ l < w

and

HNr ~ Then

E~J.

By

I~E,

So f a r w e h a v e

seen

that

,

since

such

h (~/S)

and

heIs( Z/R,

{}[6CA B Thus

xENr

Thus

I = E~N.

Ig ~

h : {{b/I,

Let

Assume

and

from

be

~(~/S)).

I ~ 0 ~ /R

follows

RECo~

SeCo~

( N / R ) ~ S ;'~. T h e n heHom( ~/R,

I=JAN

[HMT]2.3.8.

Let

CA-s.

y C I g ~ I = J.

I : E N N ~ E:J. by

Let

(VyCA) I A y ~ l < ~ .

thus

in

(ii) :

~.

Assume

Therefore

= Nr

proving

the

been proved.

HNr~ S K _C Nr~ H S K

SeCo~

F ~ Az~.

Proof

all

S~EIs((T~

extension

: z/J

(~/S).

of

a unique

we have

holds

Rgh C N r (s

Rgh = Nr

in

]7~ ~O[~IWs N L f a .

(i) h a v e

that

proves

Obviously, and

(4.1)

"if-parts"

E ENN=I

which

properly

was

it

to p r o v e

HK.

(i) :

is a u n i q u e

: J.

S

A = Dm ~

JCE.

as

rest

of

is

Then

of

repeated

for

a n d the

there

this

Nr

the

prove

be s u c h

Then

: x~IEI]

function O/S

:

By

obviously = {xEA

K

can be

B~.

KB C. C_ A

Proof

J ~ I g ~ I. If

for

Let HNr

(ii)

~CA~

ReCo~ .

to

above

hence

if-parts"

Let

Let

A=DmH

Then

~~

~

8.~9.6.

(4.1)

N r W s ~ _C I Ws a

(5) N o w w e p r o v e

Lemma

in

Again

I c s r e~ g n L f

without

some

argument

follows.

8.~8,

H(Ws N L f works

the

b/J)

that

O-ideals R = SA2N.

: bCN].

is o n e - o n e

By

by

Then

[HMT~ I = NAJ.

be arbitrary.

Let

x = c(F)x = (c(F)z)/J~

x = Thus

T~r ( ~ / S ) ) .

let 9 Nr

HNr 0[

s HS~ : Nr

~4~

(~).

since H~

By ~

by

I =

(i) w e

C 0L $31 = s

and

8.19.7.

299

QED (Lemma

Lemma

8.19.6.)

8.19.7.

(i)

(ii)

Let

~>~.

Let

~AelcA~,

Then

~(i)~,~ ~ I s ( P ~ I / F ,

Let

K _c CAB.

UpNr a

K

Nr

=

Tt d < T[~ (~i : i61>

Then

~ )

for

UpNr SUpK

and some

: Nr~ S U p K

F

be a f i l t e r

~

! P~A/F.

and

if

on

I~ a l < ~

I.

then

UpK.

c~

Proof. we

Notation:

define

~/R

the h y p o t h e s e s . = %P@[.

For

d

(R*)*~

Let

= ~(~[)

By d e f i n i t i o n ,

have

~(m),

seen

(,)

hence

of

UpNr

by

UpNr

by K

:

of

the

By the

c

proof

(A/R.

of

of

~

~

conventions

of

2(pN)AF(A)

and

Proof

[HMT32.6.32

: ]7~ PU[ EHMT3,

: F(N)Eco

ReCo61 (i) : A s s u m e

that

= m~[

and

N = Uvo~i

PT[.

Then

~7~ ( ~ / F A ) ) ,

by

PTL

=

/[ /F c and

~(N)=

by w r i t i n g

8.19.6(i).

We

SUpK

K _c CAB.

C Nr S U p S U p K

on s o m e

Nr UpK

as

with

~ P(~

~(~pU[

/FA)).

/F A c PUt/F.

Let

(,'~) we have

(PeA/F)

TU~ P U [ / F

in the Then

4j[/F,

~P[Z

(ii) :

an u l t r a f i l t e r hence

~/R

of a l g e b r a s

that

is p r o v e d Proof

c ~[

~(A):~ ~ is((TSr~ Z ) /F N,

~(A)~,: 9 I s ( P ~ o

(i)

~

notational

~

Then

/[ =9~& PUL .

By the

for

.

It is p r o v e d

C p 4A/F.

Fm

any p a i r

I. E

follows. u[elK

F

o "6/[)/F @ UpNr

K.

P("~0co ~ ) / F

above

then

P(~ao

~(PU[)

Any e l e m e n t

with

an

{ B~al<cu.

= PU[ ~

of

Nr UpK

ultrafilter

and

for

P4%/F).

Conversely,

f~.EIK

UpNr~ K _C

Suppose

Is(PTur o U[/F,

and

form

(i) we h a v e

= Nr SUpK.

Then

~(A)

By

any

F

~A)/F ~ Th6 P ~I/F c Nr

Let

F

every This

is

on

Nrc~SUpK,

of

I.

element

be

-0l C I K , implies

the

form

By the of

an u l t r a f i l t e r

above

UpNr

K

is

on

I.

UpK.

QED (Lemma 8 . 2 9 . 7 . ) (6) Then since

N O W we p r o v e K=UpK Nr

K

and ~

(iii) . Let

(V ~ N r

0.<~_<[{, and

K) I Z d J 5 1 ~ .

3x O " " "3Xn-l( A{c (<~) xi:xi

K = {4Ael Gs B : I Zd~3[ I>_~].

Therefore A Aix l #x 3

(V~UfUpNr : i<j
K) IZdlSl_>w,

: i
By

300

8. &9.8.

B_>w, 2Ws B C UfK powers Let

with

iew

: i6~> Thus

and

let

f(i)

2Ws~ _c UfK.

(i.~)

HNr

(Discussion

HNr

HSK.

SK C Nr

such SO[

# Nr

~ Nr HSK.

HSK.

(1.2)

If we d e l e t e HNr

K c Nr Let

such

that

HNr~@[

is f a l s e

for

hold (2)

HOt

~.

< f(i)/F

elements

:

~

in

Nr~ (2Ws6)

For

_~ Uf UpNr~ K

part

of

(iii),

then

~-
and

by

Then

by

HK.

~>O.

_~ N r H ~ . some

of

~_>a

and

for all to

the

some

can be d e l e t e d

By 8.19(ii)

S

as

~
K c CA B

there

is

then

statement

K c_ CAB. 8.i9(i)e

with

exists

Again

if

SK that

is n e e d e d

because

obtained

HNr such

statement

there

:3.

t h e n we of the #ACCs B

by d e l e t i n g B_>~+w

does

proves.

and

K _c CA~,

UfNr~ S U p K

this

equality

would

be

all o c c u r r e n c e s

of

candidates

(I)

UpNr

UpK = Nr

UpK

(to d e l e t e

(II)

UpNr

K

D Nr

UpK

(Up

commutes

o n e way)

(III)

UpNr

K

C Nr

UpK

(Up

commutes

the

(IV)

SUpNr c~ K

(SUp

commutes

one way)

D Nrci S U p K

and

B>~+0)

is not v a l i d

Let

#J[Ecs[egnDc B

K c CA B

this

The o b v i o u s

--

improve

is

Then

from

8.19(i)e

Hence

following.

c can be r e p l a c e d

inequality By

there

the c o n d i t i o n

Ein 8.19(ii)

This

we h a v e

of the

8.19(i)f

B>~+w

all o c c u r r e n c e

K c CA B

because

K=[eA}.

whenever

iff

all

:

by 8 1 9 ( i ) g

6>~+~,

S

8.19.)

. Let

Thus

IB~I<~

following.

not

on

Then

The o t h e r

following.

8.6.

of

6>~+~.

But

namely

S

--~ Uf UpNr~ UpPK.

be r e p l a c e d

that

here,

obtain

: jEB~w).

we h a v e

has u l t r a -

by the

ultrafilter

zero-dimensional

from

for all

MK C Nr SK

u <0

Ws B _c D i n d B

By 8.19(ii)

HNr

Nr

of d i s t i n c t

hence

elements,

nonprincipal

: jew>

and

8.19.)

8.19.8.

be

with

d < dj,i+ j

by

is n o n d i s c r e t e

zero-dimensional be any

follows

Here C cannot

8>~>O

2Ws B

Nr~ U f K --D Nr~ (2WsB)

QED(Theorem

(i)

F

But

UfNr 2 K ~ Nr 2 HSPK,

Remark

many

let

is a s y s t e m

and h e n c e

every

infinitely

O[E2Ws B

every

since

other

way)

= Nr~ SUpK. (I)-(VI) S)

below.

301

8.20.

(V)

SUpNr

(Vl)

UpNr

We p r o v e

K

c Nr S U p K

(SUp c o m m u t e s

the o t h e r way)

SK

= Nr S U p K

(Up

with

that

none

(2.1)

TO d i s p r o v e

8.6,

Ws~

of

commutes

(I)-(VI)

(V),

_c I SNra K.

let

is

valid.

B>~>I

and

By 8.8(ii),

wWsB c K c_ CAB.

By C H M T I ] 8 . 5 -

wWsa --~ U f U p N r CA B _D N r SUpK.

Hence

c~K _(~ Nr a S U p K .

SNr

(2.2)

TO d i s p r o v e

~ Nra UpK.

Let

(IV)

and

B>~+~,

Nr~U~

(II)

a>O.

By 8 . 1 9 ( i ) b

HSP

(2.3)

To d i s p r o v e

(I)

that

UpNr~ K ~ Nr

UpK

(since

UpNr

is

~J~eWs 6

such that

By 8 . 1 9 ( i ) c Choosing

(2.4)

_~ N r U p ~

it is e n o u g h

that

there

K={O'[ }

class

of

and

(Vl)

all

that

there

nondiscrete

one.

Let

obviously

In

moreover

8.19(i)e

Corollary i. (i)

8.20.

Let

Let Then Nr

CA6-s.

Then

and

~I

Let

too,

and

such

_D Nrc~ U p 6 [ .

was desired.

SK ~ Nr SUpK.

Let

cannot

be

Dind~,

the proof.

e.g.

the

UpNr

We note

following

Then

of 8 . 1 4 ( i ) - ( i i ) . )

(V~AE But

SK ~ Nr S U p K

replaced

since

with

if

HNr c~C~ : I Nr c~'~

hence

for

~<~

reg Cs6 ,

and

~J[e

U IoGS ~ .

~-
and

KE{I

Nrc~ K s = HUp Nrc~ I Cs reg,

•

Gs

and

I Gs, ~

CA, < I Crs nCA : yEOrd> IGs}. y Y '

K 6.

I Gws c~

N r HK B : HNr~ HK6. KB

,

Nr

If UpK

reg,

I Cs).

~<~+w

then

:

UpNr

KI3.

K

(V~].~UpNrct SK) IAl#a~.

is n o n d i s c r e t e } .

the p r o o f

K

B_>~+e, ~>O.

-~[ Uf Nrc~ UPJ-)[

completes

This p r o v e s

Cs B

is s i m p l e

Let

HK B = HNr

it

UpNr

counterexamples,

class

~A

KE{I Ws,

as

the proof.

a,B

K).

UpNr c ~

This

See e.g.

find

such

too.

the

l<~<w

Then (ii)

~

to

K

~J[EWsBNDc 6

completes

to show

SUpK) IBI=~.

it c a n n o t be r e p l a c e d w i t h then

enough

is

UpK D UpNr

discrete

SUpK) IBI=w.

case

K={ {][}

is

SUpNr

to s h o w that

there

UpNr K _~ Nrc~ UpK

(Hint:

(~ ~5 ENr

nondiserete

EDind B

it

K d { O [ e C A B : [ZdU[l<w

SK) IZd4Al#~.

(3)

(III)

finite

(3 ~5 eNr

CUpNr

Choosing

it is e n o u g h

But o b v i o u s l y are

.

we h a v e

TO d i s p r o v e

be the

the

Nr S).

302

8.2'1.

2.(i)

UpNr

WS6 C_ U f N r UpWs8

iff

(ii)

Uf UpNr~ WS 5 : Uf Nr~ UpWs8

(iii)

Let

KG{I Ws, I Cs reg,

Nr~ K 6 = UpNr~, K~ (iv)

Let

Proof.

iff

is a variety

(see,

e.g.,

d~&~(~5).

Let

By l(i) Then

IZd/~ I_<2

by

ICl:l.

O[ ~

For

I (ii),

first part,

: N r HK 5.

we use 8.19(ii)

The second part of

~

~

~ a d5 and

~ T ~.

for some ~ ELf B

by

EHMT32.3.14

~<~.

~SeGs 5. ~<~.

g[EUpNr~ K B _C Dind~

Then

Let

implies

by EHMTI36.14.

Let

Let

~ d

By 8.18 we have

by

~<~.

Then

that

/[

is simple

Thus using

EHMTI3

C Nr K if ~<~. Suppose ~_>~. Then 5 -- ~ 6 Nr~ K~ # UpNr~ K~, since N r K 5 _C Dind~ while UpNr~ K 6 _~ Dind~.

clearly,

(UpNrc~ K~ _~ Dind~ in part

UpNr K

can be proved

analogously

(6) of the proof of 8.19.)

2(iv)

holds of

elements. follows

Purl ~

/[ E IWs8 u IoGS 5

7.13 we have seen

members

be as in the hypotheses.

therefore

Ol ~ Then

K

then

Nr~ K8 _C K~ _C Dind~,

since

Now 2(i)

2(iv)

~

2(iii)

Vx(x:O

infinitely

from 2(iii),

2Ws 5 _c Uf K

to the proof of

By this,

reg

may contain

follows

from 2(i),

Corollary

Nr~. GwsCOmp 6

N r UpWs 6

QED(Corollary

(ii)

8.3).

class

is clear by 8.19(i).

~]t~UpNr~ KS.

(i)

Then

and the fact that each indicated

H N r HK 5 C_ H N r SHSK 5 C_ N r HSHSK~

Proof of 2(iii):

Let

reg].

Nrc~ UpWs~ ~ HSUp N r t~w~" comp~ reg

~
use 8.19(ii)

or

~<~+0~.

~,<~.

l(i),

l(ii)

iff

I Gws c~

For

again:

~I~I<~.

is proved.

V c(~,)x:l)

many

8.19(i)

and by

5>_~

zero-dimensional

and from 8.19.3.

2(ii)

and 8.19(i)d.

8.20.)

8.21.

L ~ < I K AMn

Let :

O<~
and

KE{Ws,

Cs reg,

GwsCOmp

reg,

Cs, Gs, CA}.

a~Ord>.

HSUpL

C HSPL

HSPMn

~ Ws ALf

< HSPL

: ~eOrd)

C SPCs

, and

in particular

HSUpMn

~ SP(Ws

is a system of varieties

mMn~).

definable

by a scheme

303

8.21.0.

of e q u a t i o n s , (iii)

but

La = N r L B iff

(iv)

is not

iff

(either

Conditions

HSUpL~

~2~

a.-c.

or

strongly

or

: HSUpNr K ~

below

such.

are

(Gs,

iff

L8

CA}).

equivalent.

a.

~w

B<~+~.

b.

Uf UpNrc~ L B = Uf UpNrc~ UpLB"

C. HSPNro~ LB : HSP Nrc~ UpLB" (v)

Conditions

a.-e.

below

are

equivalent.

a. d_>c0.

Uf UpL B : Uf UpR%. L B .

b.

c. HSPL~ = HSPR% L~.

Uf UpL

d.

e. (vi)

~ SUpL

c~

. c~

HSPL(~ ~ SPL . o~

Let

w<~_
and

let

Uf UpR% T = Uf UpNr

To p r o v e

Lemma

8.21,

8.21.0.

we

c~

T c UpLf B.

T = Uf UpNr

shall

Let

need

and

~,B

c(

the

L

Then

UpT.

following

lemmas.

be as in the

formulation

of

8.21,

Then

L~ _c SNr~ L~. Proof.

First

(*

I K~nMne

If

KC{csreg,

this

is not

we

_c SNr~ KB. GwsCOmp stated

it f o l l o w s

from

[HMTI]8.6,

and

if

~EK 9 (~)i.

nMn

prove

for the

then

if

nMn

K=CA

then

Then

8.21.O.)

(*)

case

[HMTI]8.5-6.

_c SNr~ (I K nMns)

QED (Lemma

reg}

If

then 6~ ~

~=i

~

holds there

KC{Ws,

(*)

holds

~

for

U[ E S N r ~ as d e s i r e d .

and

by

but

Cs, by

Gs]

by

to be p r e c i s e

GwslOmp

then

EHMT]2.6.57,

some ~

8.18(iv),

~ El K B.

E I K 6 N M n B.

reg

= Cs 1

(*) h o l d s

by

2.6.31. Let This

Hence ~

~

proves

K n

304

8 . 2 3 . . "1.

Lemma

8.21.1.

Then

Let

T ~ ~J~

Proof.

formula

e~w.

t i o n of

F

Let B

subalgebra

of

~

Let

language a~PB

q d : O{A(~i)

I

such t h a t

(Fu~)16 of

B.

be the u n i t of

of

CA -s.

: k~n}. ~ Id,

in

~Ea/F,

(Vk~n)bk/F

~.

6 : B >~

~-l*(A(ai)~g)

and let

~J[=P ~ / F

49% ~

such that

Let

~

let

be a first o r d e r

Assume

b~n+ipB

Such a

~

and

~(x,y 0 .... ,yn )

and

bki

$%.

on

be the set of i n d i c e s o c c u r r i n g

transformation

V

~eUp(Gs[egnLf~).

be an u l t r a f i l t e r

for some

Let

iEI.

Let

F

in the d i s c o u r s e

E Nr ~0~.

finite

Let

~ C I(Gs[egALf8) "

bo/F,...,bn/F]

Let

and let

is an e l e m e n t a r y

Let

for some

w~B

Then B

~ e

F C

~.

be a p e r m u t a -

and

<

is a

exists

by

~w

and

~ ~

V.

Let

rs ~ ~ < {qo~

~

E

~5i e LfB. :

&

qex}

: xCC).

EIs(~{~ ~

, ~)

:

Let

since

9 ~ <-1.

clearly B i. ors

By the p r o o f

T

Then

holds

for

Similarly, ~ Id.

every

By

kSn.

V

is a

EHMTI]8.4 GsB-unit

(VbEBi)rs~b

then

and

: sTb E B i

By

iff

~

and

e i ~ rs~a

Then

FI~ ~ Id ~i

~

.

bk/F ~ N r ~ . QED(Lemma

Then

A ( ~i)e

8.21.2.

Then

SUp'(

Proof. Let

~

where

e = < e i : i~I).

Therefore

e/F ~ Nr ~J[

Thus and

rs

: bki

o for

~-~(A(ai)~

) c

--

--

~i

~

~Eei'PJi ~

: k~n} ~ c j ( e / F ) = e / F ]

Let

Ws nMna)

Let

saw

e~w ~

and

~ Sb

w

(a)

and

and

by

Let

(Viel)

cj(e/F):e/F ~ ~

~Eai'PJi~ j~5~. _C

Ae i by

~UU

(~k~n)

~ E e / F , b o / F .... ,bn/F3.

~ Codol:X).

For any set Then

F

x=l~l>l.

Vx(at(x)

•

~J~ ~ ~ ( @ ~ a x ( O ) ) . I

~i

by

rs{(bki)

a b o v e we h ave that iff

rs{*B i -c

~i )

by

B.

(which

Thus

c ~n~

of

8.21.1.)

Lemma

Let

~Eai'PJi~

: kSn}

EHMT31.11.ZO

we have l

and by the

EA{cj(bk/F)=bklF

u{A(bki)

by

rs ~ ~ I s ( ~ i ,

thus

rs ~ E

is a p e r m u t a t i o n

rs "~Bi ~ Bi,

,. ~ reg ~oi ~s s

and

6

~iCGs[egNLfB.

i

~.

rs~V=V

Gs reg B nLf B) and by

~I~ ~ Id

Let

of

be

an

~ ~ Mn N Ws ultrafilter

on

s and I

we let ~ such

~ : < s : i<~).

is n o n d i s c r e t e . that

(VFeI)

8.21. {AeI

305

: A~F}EF.

x/F ~ dij ~(~

){x/F}.

Since

Since

H e n c e by

>~

such that

: U >~ and

QEH

U

and

and

By

= q, ~ { q }

GeH

N=Sg{y}

have

z@N3. By

~>1,

Since

~

QED (Lemma

of

proving

9 SUp{U[}

are

such t h a t

Let

~ 9

~61

Since

U=base(~).

~

Gws wa.

Then

and a c h o i c e

~

q9

and

n(Q)=q

N = Then

function

is w i d e l y

c Id

by

[ H M T I ] 3 . 1.

gEy~z.

and and

~ { ~@ ~

distributed,

feQ=G .

that

By

z~N.

prove Ay=a

that

and h e n c e

~ SUp'(Mn

n Ws

: ~

>---

: YeH]

and by

~(y)=y.

and by (Vz)

g~z

we

[O#zCy

.

T[ ~

),

that

f :

f = f-1

we have

g 9 ~{q} ~ ~(z)

y 9 At~

~

y = {ny

We h a v e p r o v e d

Let

and

Then

By

By the above,

n(G):g.

(VyEH~{Q,G})~{~yy}={n~y}

N I ~ ~ Id.

y#O

we h a v e

~

[HMTI37.16,

y ~ h(x/F).

y#O.

: YAH}.

Let

and by

This and

and by

~

H C Subb(~)

[U~(QUG)]If

therefore

~(z)#z

Let

(Vi,j 9

~5 has a c h a r a c t e r i s t i c .

Let

and

such that

fog = q.

= g

~ I{][/F.

also

Gws wd.

Then there

is a b a s e - a u t o m o r p h i s m ~{g}

and

: i<e))

be such that

foq = g

9

y = {~y

y D z # O.

are

~

Thus t h e r e a r e

I~ n a y : ~y(< nY

there

~

a cardinal

(Vi,jE~)y
{[ : uEU}.

Assume

Let

x/F > O

by 5.6(iii)

there exists

h : ~

and

n 6 P(HIId)

F.

Then

C~ has a c h a r a c t e r i s t i c

are

= Sg(~){y}

of

~ ESUpWs

[HMTI~7.26,

Then there

(VY 9

x ~ (d~ ~ ) : FEI).

by the d e f i n i t i o n

9 I G w s ~ d.

y ~

Let

Vx(at(x)

a - Codol=x)-

we are done.

8.21.2.)

N o w we

turn to the p r o o f

of 8.21.

Let

~,B,K

and

L

be as in the

hypotheses. P r o o f of

(vi):

reg Lf~ _c l~s B . =

Uf U p R d

Proof is p r o v e d b. h o l d s

T

=

of

Let

H e n c e by

Uf U p R d

(iv) :

in C a s e

(v) :

UpT

and

UpT _c UpLf B =

Uf U p N r

Suppose 2 of step

by 8.19(i ) .

P r o o f of

~<~

If Let

T ~ UpLf~.

By JAN13 we h a v e

and by 8 . 2 1 . 1 we h a v e

UpT.

O<~<~B.

Then

Nr~ Up (WsSnMn~)

(i) in the p r o o f of 8.19.

~w ~<~.

UfUpNr~ T =

t h e n b. h o l d s Then

Rd

by

If

_~ HSP N r Mn~

B<~+~

(vi).

(WssNMnB)

~ HSPMn e

since

then

8.2{.

306

Rd ( W s s A M n s )

~

{j[ C p ~ eIl(P~

with ).

Let

~IDind

.

from model under ~/J

c(~)x:c(~l)X.

E:O/F

and hence

~ $PUp ( R g ~ ) .

EHMT]2.5.25,28

by

(~) w e

: SPL

.

(vi)

L and

Mn

and

than

F

b

~][EWs

fails

claimed

of

~MnB

and by

c(•

)~ (x•

2.1.17(ii) for all Let have

Thus Nr~:B

It is w e l l

M

of a l g e b r a s P~/F

some

SPUpM

EE Assume

known

is c l o s e d

e SPUp ( ~ g ~ ) .

Thus

~<~.

Uf UpL~

) = SPUp (L N D i n d

from

EHMT]2.5.28.

Vx(Ax~l)

but

set

~ Q/F,

i in

ultrafilter. by

~w.

~ if

~ SUp' ( Ws N L

UpL

Moreover

for all

we

proved

e.g.

~ SPMn

By L e m m a

UfUpL

)

holds

~J~ is a n o n d i s c r e t e

Hence

proving

) =

T h e n b.

Vx(Ax#l) ;

EHMT]2.1.20(ii). )

Hence

) : SP (L n D i n d

Let

UpL

= L~.

# SUpL

~>i

which

for e v e r y

.

8.21.2 .

Actually,

is s t r o n g e r

nondiscrete

~ UfUpL . Let

.

~CL

~

_c ~

s

we may

Then

EHMT]2.5.25,

for

some

assume

V~ =Vs

V~Subu(~ { a~~

By

and

)} : a~

: • ~

eLB

and

~ ~GwsB

and

if

then

e<~

for all : {a•

and hence

~

V Or c ~n~.

: ~r~ ~ .

~

By

{IYI

: Ye

a~

: u{V

~N(~),

~

:

where

: ~N(~+l)}. This

By

By

proves

L~ _c

a

EHMT]

Nr~ L~

8Ae. ~w.

Nr

~

and

SUp (Ws N M n

~t ~

and

: P~/E

)

b

[HMT]2.5.25

: Ibase(V) J:~

is a f i l t e r .

L~ _c p ( L N D i n d

statement.

)) c ~ o { ~ } .

F

if

~ C o8 d o l : X )

(iii) :

8.2!.O,

Then

,

dimension

in

SUp ~

: x~E}.

T ~ Dind

follows

Uf U p L

proved

Lemma

~Subb(~

has

J=AAE

= HSP (L n D i n d

SPMn

Yx(at(x)

that

Proof

HSPL

Then

P~/F

for

Let

proved

is a n o n p r i n c i p a l

formula

the

have

~ HSP Rd LB.

J~IlO%.

Therefore

for e v e r y

we

(iii).

: i<~)/F

we have

T

= HSUpL

< doi

UfUpMn

products.

HSPL

proves

for any class

We have

T : SPUp

By

this

that

reduced

have

and : x .l= O }

theory

HSP

~IcA

F ~ {{i~I

Then

taking

~)

by

~

This

Then

K~ _C I K

= Ics[egNMn

.

Mn~ for

=

Nr~Mn B

K#Cs.

by

Thus

EHMT]2.1.17. L~ =

Nr~ L~,

By 8.18 since

and

8.1 w e

ICs~NMn~

=

8.2 2.

307

Let

~<~.

= {0 ~

,i ~

T[~ ~ E L as in

i.)

}

.

therefore Thus

~)t.

= 1~

of

~>1

follows

from

Subb(~3[)

2.)

Let

we h a v e ~Jl E G s B N M n ~

= {~,

(~+l)x{~}}.

Mn~

V-a

in

~

s

Let

[HMT32.1.23, {O,1]

HSUpMn from

formula 8.19.1-2

c(~)x=c(~l)x Hence

for

QE D ( C o r 0 1 1 a r y

~,

~

Vx(a

6.

Then

~

s CMn and

Zds .

=

By 8.18,

an,

Then

.c(~)x e

~h[ ~

by

of

~.

)

nE~

V ~ 6

Nr

be =

{a s, O}),

proving

is o b v i o u s

Vx(x. C o~ d 0 1 in

SP(Ws

for

~<e

Mn~ _c I Gs~ by

but

(Gs6NMnB)

of

Rd~ L

~

for

~1,

{O, Cod01})

NMn

).

from

(see

HSPMn the

and

~ Ws nLf

fact

is i m m e d i a t e

[HMT]

that

by

Mn

[HMT3

EHMTI]7.15.

corollary ~,

nMn

and

C SPCs

is an i m m e d i a t e

Rd H S P T

I:

is not v a l i d

MSPMn

~

~ SP(Ws Mn

for all

and p e r m u t a t i o n = HSP

Let

it f o l l o w s which

(ii)

CA].

[HMT]2.1.17(ii) LB.

By

(i):

2.1.20)

2.5.25.

by

{Gs,

.

Proof

b

K ~

= Nr

hence

HS UpMn

for

L

[HMT]2.1.17.

= -a + 1 e Nr a~

Assume

(v) by o b s e r v i n g

: L

and

since

that

HSP

for any

Rd T =

[HMT]O.5.4-13.

8.21.)

HSP

Gs

=

I Gs

:

I H 8% GiS

Uf

Rd

Uf

Nra Gs~

==?

Nra I GsB

=

Rdc~ I Gs 8

Gs

=

S

Nr

I

Gs-

Uf Up Nra Gs B H Nrc~ GsB

(~ _< ~ < 8) Figure

For hold which

any

with

~-<~<6, the

only

the

inclusions

exception

we do not k n o w

8.22.

(cf.

not

of the

Problem

indicated

inclusion

8.7).

Uf

on the Nr

Gs

figure c Nr

do not I Gs

b

308

8.2 3.

Remark

8.23

(Discussion of Figure

(i)

R % Gs B ~ Uf N r Gs 8

(2)

Let

I<~
Then

for all

Let

Nr Mn

~

Vx(Ax#i),

Nr Mn NHRd CA

(1)-(2)above.

from 8.8.

moreover

Mn

by the definition of

The rest of the inequalities and

follows

~ 6 M n . , y~a. 7 By EHMT]2.1.22 we have

~y.

every element of (3)

i<e<$

N , Ws B ~ H R ~ CAB,

for any n o n d i s c r e t e Proof.

8.22.)

About

~Df 61 ~ H R ~ CA B

b

Y

Nr

Vx(Ax#1).

Let

l<e
is discrete by 8.10.i.

in Figure

Thus Then

QED

8.22 follow from 8.15

Nr IGs B _c Uf Nr~ IGs 8

see Theorem

8.6

and Problem 8.7. (4)

The positive

statements

of Figure

8.22 hold by 8.5, 8.1,

8.19 and

by [HMTI]7.16.

We postponed one proof

from section

it uses tools developed here. implies that the algebra

~

Prop.8.24

Z ~ {q 9

Proof.

8.24.

Let

e~,

: qo ~ q'H}.

Assume

Then there is

yeA

there is

~

F c

C(F~I)Z.

such that such that

Clearly,

that

y

Since is

not regular. H~(FUAy). 9 ~ (~ (f)

)

Aylqn _9 k,

Z

is

Then and

and

f6y

H-regular

and

ITI~. q 9 ~(~

Q _c Pn-Tnqn" *T

(k)) and

and

[H~AyI~

ke~q Let

){Z}.

of

~F

~

AZ = IUH,

Since

and

be such that

is regular.

4)~Cs reg.

yESg{Z}, Let

x

such that

Let

y =

by 1.3.5 we have since

AylfCk.

Q ~ s*F. s ~ pnnqn ,

IPn*Tinlqn*T1>n,

Let

yESg(~{;F~A){X}.

by 1.3.4(ii),

be such that s ~ F1f

IHi~.

Assume

yESg(]~2F~){Z~. and hence

language

with

9 (~){Z}

is not regular.

O6F

Hence

Then

is

there.

H ~ ~i

{][ ~ ~ ( ~

Z = d(1•

H-regular. Let

Let y

be a term in the discourse = T {)[ (x).

and

~ ~ ~@~.

the hypotheses.

It

in the proof of Prop.3.7

(but not proved)

~=i~I>~

Let

section because

below fills in this gap.

constructed

indeed regular as it was claimed Proposition

3 to the present

y

Let

Let

is T

p e

AylPn ~ f,

for every

ne~.

Let

8.24.

309

new.

Then

pney

Pn' ~

Define

Then

9 a}

and

EH M T I ] 8 . 1 and

similarly

of

x,Z

By

: aEA> with

,~),

and by

: a9

and

s9

p(y)

= 9 : n9

and

Theorem

and

we o b t a i n

and

v(y)

= ~ ( ~ )(~(x)).

Fy

INI=IW~NI=~,

and

b*N

and

and

= M.

definitions

= {g 9

s 9 p(y)AF(y~M),

with

b By

ultrafilter

~ ~ << F for

y c U

the

W=base(~

and

Y

EHMTI]3.1,

) ~(x)

~.

n

(a)

: nE~>/D

:

E' ~ U

Downward that

rI(Fy)E

d p = < FYN~(a)

{g 9 same

Fy

:

goEM}

argument

N ~ W,

= {g 9

be o n e - o n e

on

some

Let =

and

h~Is( ~ ~ /D,

such

By a p p l y i n g

F(W~N)~(y), : W >~

and

algebraic

p(X)

s

: goeEn }

IYAE' I=IY~E'I=~.

s 9

Let

and by the

: goCE ' }

is

By

by

~ ECs F

By the

d= ~ F x .

s ~ Gn(Y)

by l e t t i n g

there

~

: (g~p~)E

F n (y) = T ( ~ ) (Fn(X))

thus

are

= {geFu

seF(U~E').

~ ECs F

that

= s

and

IMI=IY~MI=w.

such

bos

)

~(x)

Then

C

that

there

EH M T I ] 3 . 1 8 ,

s 9

(p(x))

Let

be a n o n p r i n c i p a l

IFI<w

and

M =d YAE'

(~)

D

--

~9

: aEA) .

(k)

d xN Ln =

and

d <{g~F Fn =

Let

S~PnCy

qn 9 e

~. "(H~F)

: go~Pne(H~F)}

~ 9 U ~ base(~

we h a v e

and

Q ~ p n * ( H N F ) n q n *(H~F)

Let

IE'INIU~E' I h w

eHo(~

by

= {geF x

: go 9

L~wenheim-Skolem


SeFn(Y)

Fn(X)

that

l~I>e~e.

: (gOq~)@a}

s ~ Pnnqn ,

EHMTI]7.3-6

such

by

En = d •

Gn e H o m ( % ~ F ~ ' ~ ) '

G n.

we h a v e

by

, ~)

Fn'

Pn E ~x (f)

(~NF)lqn,

< {g 9

Gn

for

Gn(X ) = {g 9 Then

qn' ~

since

IEnl=ILnI=~

we h a v e

_c qn ~ y"

qn ~ y'

(a~F)IPn'

~qn*(H~F). : a9

and

FW

~

: F

to

: A

: goEN}

and o n t o

such

the b a s e - i s o m o r p h i s m

~E

Is(~W,~Y) is s u c h t h a t ~ ( v ( x ) ) = p(x). Thus b ( ~ ( y ) ) = ~(T ( ~ ) (v(x)~)= (~) = 9 (p(x)) = p(y). This is a c o n t r a d i c t i o n since s e p(y) while bos

= s ~ ~(y).

QED(Proposition

To stated

save

8.24.)

space,

in S e c t i o n

some

of the

Problems

open

only.

problems

concerning

reducts

are

310

9. P r o b l e m s

Problem tions

1

Let

on

Problem

V

for

2

: KNCA

Cf.

P roble m 3

Let

finitely

generated

nl~

such that

Let

GsregnLf

Problem

-s

5

This

is n o t

Are

are

proof

of

Problem there Cf.

6

exist

3.11.

and

Cf.

Let

and

431~

(CoCl-do2)

e

there

remain

we note

"

if w e

6%

"G ~ and

Let

wording

that

IGl~x

3.7-3.9

~ 9 Cs r e g

true

replace

replace

some

~kw.

isomorphic

3.7

and

of

?

Cf.3.3-3.5.

generated

3.4-3.6.

the

EHMTI33.18)

(i')

by true

a

9 Cs r e g ~

countably Cf.

there

condition by

~ x

is the c o n d i t i o n

remains

"i~l~"

needed

"IGLg•

in

and

if w e d e l e t e

A:SgG"? the

below.

A>U{IAxI +

<x}UTUY"

~

~

replace

"IAl~x"

(i) w i t h

Are

can be generated

In p a r t i c u l a r ,

if w e

to

base-isomorphic?

condition

(iv)

and

two

base-minimal

and a

sub-base-isomorphic

not l o w e r

by replacing

3.7.

(][ E c s r e g n L f

in the p r e s e n t

connection

follows

=

and

not b a s e - i s o m o r p h i c ?

EHMTI33.18

for

ICrs~NCAa

~9

but

(VxEA)~IA•

A = SgG

if

isomorphic

~

"I~I~"

since

generated

and

(i')

a22

finitely

elements

condition

for

two

the w e a k e r

In t h i s

Is

there

Are

with

EHMTI23.18(i)c)

0.3-0.4.

a variety?

base-minimal

is i m p l i c i t

condi-

1.6.

are

which

and necessary

< 2

"IAe~x"

(which

Cf.

KNCA

~ ~

saw>•

Does

?

sufficient

O < CoC2-do1

~kw.

-s w h i c h

4

Is

the

I c r s Z~d r e g , I crslreg}.~

eg,

then

and

csregnLf

Problem

{I Crs

~ HCrs O r e g

< i

0.9

are

v) 9 C A

: HK?

Crs

b O < CoCl-dol 9 Zd~5.

K 9

ICrs

that

What

T~(~@

Let or

Note

V c ~U.

with

: xEA}.

"G { <x}UT"

in the

EH M T I ] 3 . 1 4 .

~,

~

E

Cs reg

ext-base-isomorphic

and

let

to b o t h

U[ ~ ~ 0[ a n d

.

Does

d5

?

311

Problem

7

For

any

Crs

~A ,

: iEUF>,~)

-choice

function

introduced

in Def.

3.12.

Let

~Jl,~ E Cs

an

(F,
with

8

Let

in P r o b l e m _
elements}, By

(~)

Let

q+(~)=~+l

These

motivate

Is

rq+(~)

q

for

udAFEHO~)[

there c

< F,< b a s e ( % % )

and

a

UddD6ISfS?

: HxH ~ w

and

: every

6K

: rq(~,B)=l} for all

~eH~5,

and

~+i ~ q+(~)

) :

3.10-3.13.

q+

" H ~ w

for all

for

and

B , UddD:~

b e as

~ d(~x~)<

can be generated

@eH.

[ l o g 2 ( 1 6 ~ ( a + l ) I+2)]

F,D


Cf.

:

was

ultrafilters

A ~.. UdcF'fJ[

that

d G{nee

d u{6~

~

and

BK @ d BCs A M d { W x ( ~ ( @ x @ ) _ > x > O

= @+i+I~{4]I

~ q(~,B)

Hence

Let

Are

function

such

Let

rq(e,B)

rq+(@)

rq(~,8)

.

UdcFeISU[,

[HMTI3.

and

[p3],

d

H d w~2.

2 of

: i6~}.

and

F

the h o m o m o r p h i s m

0[ ~ ~

function

base-isomorphic

Problem

c

: ieuF>,~)-choice

: iEUD>,@)-choice are

ultrafilter

by

n

~,~EH.

The

authors

all

~,BeH.

~ ~+2

proved

for all

~6H.

the questions:

q+(~)=~+2

for

some

~EH?

Is

q+(5)=~+2?

Is

q:rq?

Is t h e r e

an

+ approximation In t h i s concerning

of

q

better

connection the

above

we

than

note

problem

(~)? A r e

that

Ep3~

as w e l l

q

and

contains

as P r o b l e m

q

monotonic?

several 2 of

results

EHMTI3.

Cf. EHMTI]

4.5-4.8.

Problem c

I Cs

9

?

Let Cf.

Problem

~A~

Is

{ 0[E Gs n D c

:

Subb(~)

l<w

i<~<~}

and

c

4.14-4.18.

i0

Is e v e r y

epimorphism

surjective

in

and

CA

in

Gs

?

Cf.5.11. The

Gs - p a r t

whether

to

~

or

the

variables

property 9

of t h i s

not

(By

of P r o b l e m logic

Lt c F

and with

the

restricted

of

is

is e q u i v a l e n t

introduced

t = <~

notations

logic

10 a b o v e

Fr

the

in

I-AGN23p.36 when

: iEw )

has

quoted

paper

~ ,

and

to the q u e s t i o n

the the

the c l a s s

Beth set

of

restricted

definability of

all

formulas

its models is

312

~

: i~>,

Problem

11

question look

Problem

if

~.

on Fig.5.10

Let

of

K

reg

Problem

wSdind

13

Let

CA - s .

or

c IGws c~

that

Note

settle

the

xny#O3.

Problem 6ZLE Cf.

NCs

K

the

Cf.

6.6

and

a~m. reg,

in

since

6.3-6.6,

Let

l<~<~e

.

Is t h e n

2 ~

indicated

[HMTI]5.7,

by

6.9,

6,10

Cs

Gs,

?

How will

Gws}

figure?

Fig. E H M T I 3 6 . 9

we r e p l a c e

Is

HPCs reg

all

=

EHMTI]6.8.

Is

of all w e a k l y

wSdind

or

wSdind

the

proof

wSdind

nGws

subdirectly

c IGws c~

nGws c~

reg,

reg c Icsreg?

of

5.6(i)

nGws c ~

reg

does

not

N ~xVy

work

to

[(Ax)nAy=O

~

EH M T I ] 6 . 2 3 - 1 6 .

and

K C

c UfUpK

?

{Ws, Is

csreg,

GwsCOmp

U f U p () Gs_ n L f

reg}.

_ '~ U f U p (

Let csregnLf

)?

7.4-7.7.

Problem

15

Does

questionmarks

Problem

16

Uf Up ooWs ~

where

Let

(inclusions)

indicated

by

hold?

~_>~.

Which

ones

of t h e

) = Uf UpGwsCOmp~ reg,

17

was

Let

~ ~ 2 I~oBI

conditions

Problem

of t h e e q u a l i t i e s

following

equalities

are

true?

= S Up ooWs ~ , Uf Up ~ Cs reg = Uf Up W S Ul oCs , Uf Up ~ CS r e g = I coCs

wSdind

Problem

some

on Fig.7.6

Uf Up ( D i n d N G w s

Does

in t h a t

Let

14

Cs r e g ,

be t h e c l a s s

question 5.6,

Ws,

= HP

K

construction

last

How do Figures

I Gs

{Cs,

5.6,

indecomposable wSdind

Is

with

Cf.

equalities

is i n c l u d e d ?

K E

= HPGwsCOmp reg? C~

s o m e of t h e

hold?

2_<~<~_<~.

if f o r all

occurrences

~.)

Does

iGwsCOmp

12

like

any ordinal

Let

marks

like

look

for

18

of

defined

x~B

infinite

~Cs~ e g ~ I •

7.14(2),(3)

For which

in P r o b l e m

be two

imply

(wSdind nGwsCOmp

needed?

B>~>2

is

Cf.

13.

Cf.

reg)

_c Uf U p C s reg

7.6-7.7.

cardinals.

Is

How much

~Cs ~ _c I ~ Cs 9 ~ are

the cardinality

7.14-17.

Nr~ I Gs B = Ul

Nra G S B ?

Cf.

8.6.

313

Problem

19

For which

Problem

20

Let

cardinal"

then

the p r o o f

of

E

6>~_>~.

If

is

~ >_

8%

we

(cs[egnLfs)

"the

UIUpWsc~ = UfUpRd Wsg.

8.13

{Ws, Cs r e g , +

151>2 I~1

first

Is

c ICs

uncountable

this

?

measurable

condition

necessary?

161:1~1 ~- (UfUpKa = UfUpRd~ KB)

have

WsC~Lf, c s r e g m D c } ,

so the

question

Cf.8.10(4).

concerns

for

the

By

K e

case

6>I~I

Problem

21

By

Uf Up (Cs nLf

EAN8]

and the

) : Uf Up (Cs NDc

= Uf Rd~ Uf Up (CsBNDc6)

for

proof

of

8.13

and

8.4 we

) : Uf UpR% (CsBNLfs) 6~eAw.

Can

Cs

have

=

be r e p l a c e d

with

Ws

or

Cs reg here? Note

that

it can be r e p l a c e d

CraxUtr~{CO-C7}

Problem and

22

in the p r o o f

Does

8.13.4

~

:

What

is the

Problem ADcB)

23

Cf.

of

with

for

free

nCw

8.13,

8.13

~ Ccs~eg}?

: m
e.g. by

using 8.21.

generalize

to

ICs reg

Let

: ~(i,v)

is a f o r m u l a

in the d i s c o u r s e

variables

language

of

i,V,Xm,Um, Sm,Jm,m
(Sl)-(S9)OEx?

For which

B>~

is

Rd (csBnDcs)

c ICs

reg~ H

R% (US8

or

_c ICs~?

Problem 8.147

24

To w h a t

How much

uncountable IBi>~>~, that

answer

8.13.

3zVi~(i,ext(z,i))3

~(i,V,Xm,Um, Sm,Jm Crs-structures

of

CA,

or the p r o o f

(Sl)-(S9)U@p{str(~)

Ex ~ { E V i 3 ! v ~ ( i , v )

with

is the

measurable by the

proof

the c o n d i t i o n

Problem

Uf UpLf 5

25

extent

cardinal method

l~i:w

Dc B

the c a r d i n a l i t y

condition

In 8.21(vi)

nor w i t h

are

if

I~I=~

needed?

implies

of

8.13

Ws

but

can

be o m i t t e d

UfLf 5

cannot

6~+(~

(by

conditions The

existence

~ U f R d Ws B

for

it is c o n s i s t e n t from

some

with

Then

neither what

are

in

of an

8.~4.

be r e p l a c e d EAN83).

needed

with the

ZFC

314

necessary conditions? Problem 26

In 8.14(iii)

the new condition

the condition

"lel=w '' can be replaced with

"there is no uncountable measurable cardinal

~ e".

Is this new condition necessary? Problem 27

Let

= SNrK ~ ?

K9

Cf. 8.18.1

Problem 28

Let

reg

such that

Problem 29

Let

Let

Let

8>~>~.

Is

K

=

~EGwsreg.

Does there exist

~ C

rs~ 9 I s ( ~ , ~ ) ?

6>~>O.

is regular in

reg}.

and 8.18(iii)

8>~>O.

eSNr~ uws B

(VxeB)[x

Cs reg, IGws c~

Let ~

3.

~eGws B Is then

and ~( ~)B

~

~

Assume

%UL.

regular?

REFERENCES

[A~ Andr~ka,H., Universal Algebraic Logic, Dissertation, Sci.Budapest 1975. (In Hungarian)

Hungar. Acad.

[AGN13 Andr~ka,H. Gergely,T. and N~meti,I., Purely algebraical construction of first order loqics, Publications of Central Res.Inst. for Physics, Hungar. Acad. Sci., No KFKI-73-71, Budapest, 1973. [AGN23 Andr~ka,H. G e r g e l y , T and N~meti,I., On universal algebraic constructions of logics, Studia Logica XXXVI, i-2(1977), pp.9-47. [ANI3 Andr~ka,H. and N~meti,I., A simple, purely algebraic proof of the completeness of some first order logics, Algebra Universalis 5 (1975), pp.8-15. EAN23 Andr~ka,H. and N~meti,I., On universal algebraic logic and cylindric algebras, Bulletin of the Section of Logic, Vol.7, No.4 (Wroclaw, Dec. 1978), pp.152-158. JAN33 1978.

Andr~ka,H.

and N~meti,I.,

O_~nuniversal algebraic

logic,Preprint,

EAN43 Andr~Ka,H. and N~meti,I., The class of neat-reducts of cylindric al~ebras is not a variety but is c~losed W.r.t. HP, Math. Inst. Hungar. Acad. Sci., Preprint NO 14/1979, Budapest 1979 ~. Submitted to The J. of Symb. Logic. [AN5~ Andr~ka,H. and N~meti,I., Neat reducts of varieties, Math. Hungar. 13(1978), pp.47-51.

Studia Sci.

[AN6~ Andr~ka,H. and N~meti,I., ICrs is a variety and ICrs reg is a quasivariety but not a variety , Preprint Math. Inst. Hungar. Acad. Sci.,

315

Budapest,

1980.

JAN73 Andr~ka,H. and N~meti,I., On the number of generators of cylindric algebras, Preprint, Math. Inst. Hungar. Acad. Sci., Budapest 1979. [AN83 Andr~ka,H. and N~meti,I., Dimension complemented and locally finite dimensional cylindric algebras a r e elementarily equivalent, Algebra Universalis, to appear. JAN93 Andr~ka,H. and N~meti,I., Varieties definable by schemes of equations, Algebra Universalis Ii(1980), pp. 105-116. [CK3

Chang,C.C.

and Keisler,H.J., Model Theory, North-Holland,

1973.

[G3 Gergely,T., Algebraic representation of language hierarchies, Working Paper, Research Inst. for Applied Computer Sciences (Hungary), Budapest 1981. To appear in Acta Cybernetica. [H3 Hausdorff,F., Uber Zwei S~tze yon G. Fichtenholz und L. Kantorovitch, Studia Math. 6(1936), pp. i8-19. [HMT3 Henkin,L. Monk,J.D. North-Holland, 1971.

and Tarski,A.,

Cylindric Algebras Part I,

[HMTI3 Henkin,L. Monk. J.D. and Tarski,A., Cylindric set algebras and related structures, this volume. EM3

Monk,J.D., Mathematical Logic, Springer Verlag,

1976.

[M13 Monk,J.D., Nonfinitizability of classes of representable c[lindric algebras, The J. Symb. Logic 34(1969), pp. 331-343. [N3 N~meti,I., Connections between cylindric algebras and initial algebra semantics of CF languages, In: Mathematical Logic in Computer Science (D~m~iki,B. Gergely,T. eds.) Colloq.Math. Soc. J.Bolyai Voi.26, North-Holland, 1981, pp. 561-606. [N13 N~meti,I., Some constructions of cylindric algebra theory applied to d~namic algebras Of programs, Comput. Linguist. Comput. Lang. (Budapest) VoI.XIV(1980), pp.43-65. [p3~ P~Ify,P.P., On the chromatic number of certain graphs, Math. Inst. Hungar. Acad. Sci. Preprint NO 17/1980. To appear in Discrete Mathematics. [$3

Sikorski,R.,

Boolean Algebras,

Springer Verlag,

1960.

317

INDEX

This book

list

EHMT].

should For

be used

EHMT]

OF

SYMBOLS

together

with

the index of symbols

see any one of the lists

of r e f e r e n c e s

in the

in this

volume.

~x,

Ax

[iE~

AEV]x,

A(U)x

: c l x#x}

dimension

2, 133,

;

set of

x;

EHMT]

132-3

fxu

{< x , u > } U ( D o f ~ { ~ } ) I f

f()~lu)

f•u ;

4

fEH/g]

H1g U

(DofNH)If

~U

set of f u n c t i o n s

from m to U;

~u(P)

{qE~U

;

DEV]

diagonal

cEV]

cylindrification;

Crs

, Cse,

Ws

, Gs

, Gws

: lq~pl<~}

4

132 1,5,EHMT]

5, [HMT] 4, E HMT]

element;

4,

E HMT]

d i s t i n g u i s h e d c l a s s e s of c y l i n d r i c - r e l a t i v i z e d set algebras; 5-6

Crs~egt etc.

K reg,

;

;

class

of r e g u l a r

members

of

K;

6

K,

Gws,

etc.

class of m e m b e r s of K w i t h all s u b b a s e s infinite; iO6(I.7.20), 134, 72

K,

Gws,

etc.

{~eK

: (VUCSubb(~))IU]=~}

; K;

138(0.5)

of

K; 138(O.5)

K n~

Gws n~

etc.

class

of normal

K c~

Gws~ ~

etc.

class

of c o m p r e s s e d

K wd,

Gws~d, etc.

GwsCOmp Koreg,

reg Kzdreg,

Kcreg,

Kireg

(Gws~~

6 together

reg

( xNW

: xCA)

see

r~w {~

RIw{~ , RIwA, ~W6~,

RI(W)A

~d(W)[~ , ~[(W)A

;

members with

of

138

152(1.6.1)

rs W rl(W>

members

distributed

{~b ~

rlw,

of

class of w i d e l y K; 138(O.5)

Rl K, Rs K

rl~w , rl A, rlA(w),

members

134

: ~CK,

universe

; of

~(~W)rlw;'~A

beA}

;

;

6

73(I.6.1) 153(2.1)

P/W[~; ;

base-isomorphism 155(3.1)

153(2.1(ii))

153(2.1(ii)) induced

by f; 37(I.3.5),

318

v~


: x is a set>

ANB + c

{aeA

: a~B}

;

;

[HMT3

[HMT3

d e f i n e d if c is an ( F , U , ~ ) - c h o i c e tion; 86(1.7.1)

Rep(c),

Rep(F,U,e,A,c),

Rep

r e p r e s e n t i n g f u n c t i o n of u l t r a p r o d u c t s of Crs -s; 86(I.7.1) RePc

Rep c A ud F , ud F

~ Rep(c)

;

244,

diagonal ultrapower 162(3.5.1)

onto

function;

sub-base-isomorphism;

132

>~

one-one

function;

>~

one-one

and o n t o

c

"finite

subset

V

{X

: X ~cW

Sb V

powerset

~

full

v

universe

Zd

{xeA

the

of V;

of

132

relation;

132

132 [HMT3 unit

V; of

132

132

;

133,

~ ) Zd~3% ; 133,

corresponding 133

132 4)t ;

~(~);

: A(~)x=O}

~(~O etc.

function;

of"

;

with

132

subalgebra

Mn(~)

AtA,

V}

Crs

minimal

ZdA,

86

d i a g o n a l u l t r a p o w e r h o m o m o r p h i s m if c is an
udA F , ud c

Sb

262-3

262-3

notions

for

base(V),

base(~)

base

Subu(V),

Subu(~)

set of

subunits

of;

133(O.1)

Subb(V),

Subb(~)

set of

subbases

of;

133(O.1)

<s Sm ~ I H

Sm ,

of;

: iE~)

~([@UA)A;

133(O.1)

;

set of small

141 elements

of

~

;

146(1.2)

146(1.3.1)

I H

Dm H ,

Dm H ,

DmH('UI-)

{xeA

~1' H ,

~H'

~&H (~)

s u b a l g e b r a of 146(1.3.1)

H-dim ~

func-

H-dim

: IAxNHJ<~}

;

146(1.3.1)

4)[ w i t h

universe

set of a l m o s t H - d i m e n s i o n a l 0[; 195(4.7.2.1)

DmH;

elements

of

319

fAB

base-relation

Ud K

class of d i r e c t e d 203

Uf K

c l a s s of u l t r a r o o t s (7.0), [HMT]

Up 'K

c l a s s of u l t r a p o w e r s 229 (7.O), [HMT]

Dind

{ @t6CA

rb O , r b (p)

rb(P)

d rb p ;

191(4.7.1.1)

rd p , rd (p)

rd(P)

d rd p ;

191(4.7.1.1)

rd~

rd (~IId)

rs

289 (8.16

Rd ~(P) BoB

263

Ord

c l a s s of all o r d i n a l s ;

CAH,

CSH,

etc.

;

Nr H ~Jt,

d e f i n e d for [HMT]

CA,

I

Cs,

etc.

(Si)

- ($9)

Crax,

Cpax,

of m e m b e r s

of m e m b e r s

170 of K;

of K;

of m e m b e r s

of

229

K;

210

261

H-dimensional defined

~(~)

unions

: IZd~l_<2};

Rd H tO[,, q{~H~Jl T ~ H 6[

i n d u c e d by f on AxB;

for

CA-s;

263

222(6.0)

4JIECAs, SDH; 6~6CAs,

greatest

csregNLf

systems

of c l a s s e s

222(6.0), [HMT]

SD_H;

269, 222 (6.0),

with base

~;

230(7.1)

of a l g e b r a s ;

263(8.2)

281(8.13.1) Rgax

sets of axioms;

str (~)i)

Crs-structure

Cyl~ ( ~ )

Crs

Id

identity

281(8.13.2) associated

associated

to

9]~;

r e lation;

Do R ,

DoR

domain

Rg R ,

RgR

r a n g e of R;

of R;

to .0[ ; 281(8.13.3) 281(8.13.3)

[HMT]

[HMT] [HMT]

A]R

R domain-restricted

R~'~A

R-image

of A;

R*x

R~':{x} ;

[HMT]

PJa

a-th projection;

Op K

t h e o r y of K;

Md~p

c l a s s of m o d e l s of 9;

to A;

[HMT]

[HMT]

[HMT]

[HMT] [HMT]

320

Su~% $K,

set of sot

class

subuniverses of

of

subalgebras;

Sg(~)x,

SgX

subuniverse

~ ( 0 [ ) x,

~X

subalgebra

of ~ g e n e r a t e d of

~

generated

class

of h o m o m o r p h i s m s

IsUt

class

of

h * U[

h-image

HO ( (~ , ~5 )

set of h o m o m o r p h i s m s

Is(~,

set of i s o m o r p h i s m s

of

~

set of

EHMT~

6% ; CHMT]

from ~ image

class

of h o m o m o r p h i c

I K,

class

of

set of c o n g r u e n c e

Ii~t

set of

ideals

of

generated

of

images;

; [HMT] ; [HMT3 ; [}{MT]

~ ; [HMT]

~

; [HMT]

[HMT] [HMT]

relations

on

t/L ; [HMT3

tit ; [HMT] by X;

direct

PK

class of i s o m o r p h i c p r o d u c t s ; [HMT3

images

of d i r e c t

c l a s s of i s o m o r p h i c products; EHMT]

images

of u l t r a -

ur

K

cr~

~5 is a s u b r e d u c t

K

reduct

union

< A,+,-,-,O,1)

Boolean

BA

class

E(~), E

sup;

[HMT]

H(~)

inf;

[HMT]

t

< A,+,.,-,O,I,c

,dxk >~,I<~

)t

dual

~

; [HMT]

of

of K;

algebra;

[HMT]

algebra;

; 263,

fJL ; [HMT3

E HMT3

of all B o o l e a n

cylindric ~O/A

C

of

[HMT3

P~s ' PieI ~i

Up

product

into

images;

isomorphic

Co~A

onto ~

from ~ into ~

isomorphisms

HK, HC~L

ideal

by X;

from ~ onto ~

~J[ is a h o m o m o r p h i c

Ig X

[HMT~

~J~; [HMT]

from ~

Ism(

Ig(~[)X,

on

by X;

; [HMT3

set of h o m o m o r p h i s m s

IUL

on

isomorphisms

Hom(Ot , ~ ) 4][ , ,~ )

; [HMT]

[HMT]

Ho 4A

~5)

~

algebras;

[HMT]

EHMT]

[HMT3

cylindrification;

[HMT~

)/

sx

substitution

operation,

k for

x;

[HMT]

321

s(~,i)

clF 6%,

s

substitution operation, and ~; EHMT]

interchanging

BA of

of

F-closed

elements

4)t ; E HMT]

c(F)

generalized

cylindrification;

EHMT~

dr

generalized

diagonal

EHMT]

~R

generalized

co-diagonal

Lf

class

mc

c l a s s of all d i m e n s i o n - c o m p l e m e n t e d [HMT3

Mn

class

of all

locally

of all m i n i m a l

element;

finite

CA -s;

class

of

reducts;

Nr K

class

of n e a t - r e d u c t s ;

EHMT]

Bo

class

of all B A - s

operators;

At Ut

set of a t o m s symmetric

RIS

a~

of

r a

c(~)d(~x•

at(x)

formula;

mr

Rs

!d P~ ker(f}

CA

a

EHMT]

with ~

; p.225

of

EHMT3

EHMT]

5HMT]

of the

relations

234(7.3.1),

R and

[HMT]

234(7.3.1) ; EHMT

is a s u b d i r e c t {< x,y}

CA -s;

[HMT]

difference;

relative product S; [HMT]

[HMTI

EHMT]

of

Rd K

; 263,

EHMT]

CA -s;

p-reduct

Rd (P)K,

~

element;

product

of

~;

EHMT3

: 3z(<x,z)Ef and Ef)} ; 238

322

INDEX

OF

DEFI NED

TERMS

This list should be used together with the "index of names and subjects" of the book [HMT3.

base, 4,5,133 base-isomorphic, 37 base-isomorphism, 155 ext-base-isomorphism, 156 lower base-isomorphism, 156 sub-base-isomorphism, 156 base-minimal, 157 base-relation, 170 Cartesian space, 5 weak Cartesian space, 5 choice function: (F,U,~)-choice function, 86 compressed Gws, 17 or Crs, 138 cregular, 152 Crs-structure, 280 cylindric field of sets, 5 cylindric-relativized field of sets, 4 cylindric-relativized set algebra (Crs), 4 cylindric set algebra (Cs), 5 definable by a scheme of equations, 263 strongly definable by a scheme of equations, 263 ext-base-isomorphic, 156 ext-base-isomorphism, 156 strong ext-(base-)isomorphism, ext-isomorphic, 45 ext-isomorphism, 155

156

field of sets (cylindric-relativized, cylindric, generalized cylindric, weak cylindric, generalized weak cylindric), 4-6 (F,U,~)-choice function, 86 generalized generalized generalized generalized

cylindric field of sets, 5 cylindric set algebra (Gs), 5 weak cylindric field of sets, 6 weak cylindric set algebra (Gws), 6

hereditarily disjoint, 199 H-regular (for any H ~ ) , 145 i-finite, 153 iregular, 152 irreversibly small, i-small, 197

197

~-regular ultrafilter,

98

lower base-isomorphic, 156 lower base-isomorphism, 156 normal Gws,

17,

or Crs,

(q,F,K)-small, 197 Q-weakly small, 189 Q-wsmall, 189

138

323

regular, 6, 145 regular in V (or in /A), 145 H-regular (for any HOe), 145 zdregular, iregular,--cregular, 152 regular cylindric-relativized field of sets, 6 regular cylindric-relativized set algebra (crsreg), 6 regular field of sets (cylindric-relativized, cylindric, generalized cylindric, weak cylindric, generalized weak cylindric), 4-6 regular set algebra (cylindric-relativized, cylindric, generalized cylindric, weak cylindric, generalized weak cylindric), 4-6 regular ultrafilter on a set I (= III-regular~, 98 x-regular ultrafilter, 98 relativization, 12, 153, [HMT3 residually nonzero characteristic ( ~ is of), 157 set algebra (cylindric-relativized, cylindric, generalized weak cylindric, weak generalized cylindric), 4-6 small (in ~ or in V), 145-6 irreversibly small, 197 i-small, 197 (q,F,K)-small, 197 Q-weakly small, 189 Q-wsmall, 189 weakly small, 190 wsmall, 190 strong ext-base-isomorphism, 156 strong ext-isomorphism, 155 strongly definable by a scheme of equations, 263 strongly ext-(base-)isomorphic, 156 strongly sub-(base-)isomorphic, 156 strong sub-base-isomorphism, 156 strong sub-isomorphism, 156 subbase, 5, 6, 133 sub-base-isomorphic, 156 sub-base-isomorphism, 156 sub-isomorphic, 45 sub-isomorphism, 156 subunit, 133 system of classes (of algebras), 263 weak Cartesian space, 5 weak cylindric field of sets, 5 weak cylindric set algebra (Ws), 5 weakly small, 190 widely distributed Gws, 17, or Crs, wsmall, 190 Q-wsmall, 189 zdregular, 152 z~imensional

closure,

135

138

cylindriG,