Band generated by two semilattices is regular

Semigroup Forum Vol. 20 (1980) 335-341 RESEARCH ARTICLE

A BAND GENERATED BY TWO SEMILATTICES IS REGULAR Peter R Jones Communicated by J. M. Howie In [2] P. Olin described the free product, in the variety B of bands, of two chains, deducing as a corollary its regularity. band is regular if it satisfies the identity

(A

axaya = axya).

By

proving the theorem stated in the title (Theorem 5) we generalize this result.

In addition the B-free product of two semilattices is

described as the spined product of their free products in the varieties

RR

and

LR, of right and left regular bands respectively,

normal forms for the elements of these products having been provided in [2]. It is of interest to note that our main theorem and that of Olin provide alternative proofs of the result of H.E. Scheiblich that the band of idempotents of the "free elementary *-orthodox semigroup" (see [3]) is regular (for as noted by T.E. Hall, this band is freely generated by two Let B Any element

e-chains).

be a band generated by two semilattices b

of

B

can be expressed in the form

the terms belonging alternatively to expression for

b

U

and to

U

and

V.

of least length reduced, denoting by

corresponding number of terms.

That

B

V.

b = b I ... b , n Call any such IbI

the

satisfies the identity

axaya = axya will be proved by induction on Ixl, IYl and lal.

It is convenient

to prove two preliminary lemmas on bands in general.

The first is

immediate from the fact that on any band Green's relation

J

is a

congruence whose classes are rectangular bands.(See [i, Chapter IV]). 335

0037-1912/8~002~0335 $01.40 f~1980 SDnnger-VerlagNew YorkInc.

JONES

LEMMA

i.

contains If

s

LEMMA

Any

all p r o d u c t s

and

2.

finitely

t

If

generated involving

band

S

every member

are two such p r o d u c t s

a

and

sasb = s(ab)sb

b

then

are c o m m u t i n g

for all

has a k e r n e l w h i c h of the ~ e n e r a t i n g

sut = st

elements

of a band

S

then

by s e m i l a t t i c e s

U

and

s 6 S.

sasb

F r o m n o w on

LEMMA

3.

both

a

first

We may suppose

=

(sasb)a

=

sasabsasb

=

sabsasb

=

sbasasb

=

sbasb

=

sabsb

is a band

a

that

a 6 U.

is satisfied.

and

N o w let

k >

2

y

a, x

belong

if

Otherwise

and

.

is s a t i s f i e d

and

Clearly

axaya

(sasb)

generated

a x a y a = axya

x, or both

Suppose

identity

B

The i d e n t i t y and

PROOF.

y x

t o_o U

or

y

=

(axaya)yx

=

axayxaya

=

axaxyaya

=

axya

the i d e n t i t y

B

with

a, x 6 U U V

and

IYl < k

B

with

a, y 6 U u V

and

Ixl < k.

y = yl...y k a reduced a 6 U

and, If

as above,

Yk 6 U

expression

and for y.

U to

or U

V. the

By L e m m a

i,

for all

and for all Let

a, x, y

lyl = k, w i t h We m a y again

then =

axay I ... Y k _ l Y k a

=

axay I ... Y k _ l a Y k

336

to

to V.

x 6 V.

axaya

V.

is satisfied

in

a, x 6 U U V

whenever

.

in

that

B

V.

(axaya)

a, x, y

be such

or

belongs

x and y b e l o n g

suppose

in

all b e l o n g

a, x, y B

u 6 S.

F r o m Lerm~a i w e have

PROOF.

in

for all

set.

suppose

JONES

Otherwise with

a.

axy I ... Yk_laYk

(since

=

axy I ... Yk_lYk a

=

whence

and therefore

Yk E V,

Consider

the p r o d u c t

From Lemma i, (axaya)(axya)

Now

YkXY

(where if

Yk-i ~ U

Thus

. commutes

(axaya)(axya). (axay)(xya)

=

(axay)(Ykxya)

=

(axay)a(Ykxya) .

=

Yk(XYl...Yk_2)(Yk_l)Yk

k = 2, x Y l . . . y k _ 2 = x)

since

axya

=

= by hypothesis,

lyl...Yk_iI
=

,

,

Yk(XYl...Yk_2)Yk(Yk_l)Yk IxYl...Yk_21

,

Yk-i E U

and

< k.

Ykxya

=

Y k ( X Y l . . . Y k _ 2 ) [ y k ( Y k _ l ) Y k a]

=

Y k ( X Y l . . . Y k _ 2 ) [ y k ( Y k _ l a ) Y k a]

,

using Lemma 2,

= reversing

the step in the previous

whence

by hypothesis,

aYkxya

by hypothesis,

=

a(YkX)a(Yl...Yk_l)aYka and

Yk E V

(axaya)(axya)

and so, using L e m m a i again,

Thus

axaya L axya

(since

a(YkX)(Yl...yk_l)aYka

since yk x E V

since

paragraph

=

=

Hence

Yk(XYl ...Yk_2 ) (Yk_la)Yk a ,

lYl...Yk_II

aYkxa(Yl...Yk)a and

lYl...Yk_ll

=

(axay)aYkxaya,

=

(axay)(Ykxaya)

=

(axay)(xaya)

=

(axay)a(xaya)

=

axaya.

Yk_l a E U),

,

< k , = aYkxaya , < k .

.

On the other hand consider

the product

337

(axya)(axaya).

JONES

Note first that if

Yl 6 V axya

then

=

a(xYl)(Y2...Yk)a ,

where

=

a(xYl)a(Y2...Yk)a

=

a(x)a(Yl)a(Y2...Yk)a

=

axaYl(Y2...Yk)a = axaya

xy I e V,

(using the inductive hypothesis at each step). Otherwise

Yl 6 U

(axya)(axaya)

and =

(axy)(xaya),

by Lemma i,

=

a[x(Yl)(Y2...Yk)x]aya

=

ax(Yl)X(Y2...Yk)xaya

=

ax(aYl)X(Y2...Yk)xaya

,

by the dual of Lemma 2, =

ax(aYl)(Y2...Yk)xaya,

reversing the previous step, since

Thus

ay I 6 U,

=

(axay)(xaya)

=

(axay)a(xaya) = axaya

axaya R axya, so

.

axaya = axya.

The identity is therefore satisfied whenever IyI = k, and similarly whenever

a, y 6 U U V

a,x E U U V and

and

Ixl = k.

The

lemma then follows by induction.

LEMMA 4. a6

U

The identity

or

PROOF. Ixl = I

is satisfied in

B

whenever

V.

By the previous lemma, the identity is satisfied whenever or

IYI = I.

satisfied whenever with

axaya = axya

Ixl +

k > 2

a 6 U u V

IyI = k.

reduced form.

Let

and

We may suppose

and suppose the identity is Ixl +

lyI < k.

Let

IxI > i, x = Xl...Xn,

Then axya

=

a(x l)(x2...xny)a

=

a(xl)a(x2...XnY)a ,

=

axla(x 2. ..Xn)a(y)a,

338

by Lemma 3,

x, y E S, say, in

JONES by hypothesis, =

a(xl)(X2...Xnay)a,

=

axaya .

by Lemma 3,

The lemma now follows by induction.

Finally we show that the identity for all

a, x, y

in

B,

lal = 1

has been proved.

is satisfied for all k = la[.

a

axaya = axya

by induction on Let with

k > 2

lal.

is satisfied

The case

and suppose the identity

[a I < k.

Let

a = al...a k

with

Then axaya

= (al...ak)x(al...ak)Y(al...ak) =

(al...ak_l)[ak(xal...ak_l)ak(Yal...ak_l)ak]

= (al...ak_l)[ak(xal...ak_l)(Yal...ak_l)ak] =

[al...ak_l(akx)al...ak_l(Y)al...ak_l]a k

=

[al...ak_l~kX)(Y)al...ak_l]a k

= axya . This now completes the proof of our main theorem. THEOREM 5.

Any band generated by two semilattices i_ssregular.

In particular, of course, the free product, in the variety of bands, of two semilattices is regular.

This generalizes

Theorem 1.1 of [2] where the statement was proved for two chains (considered as semilattices).

In [2], P. Olin showed (Corollary

i.i) that Theorem 5 is best possible,

in the sense that the free

product (in the variety of bands) of two semilattices, at least one of which is non-trivial,

generates the entire variety of

regular bands. Olin proved his Theorem i.I by first giving a set of canonical words, together with a product, for the free product of two chains. constructive.

Our proof of Theorem 5 is of course nonHowever the result itself can be used to describe

the free product of two semilattices,

in a manner slightly

different from that of Olin, as follows. 339

JONES

If U *V V

V

is a variety of bands and

their

V-free product.

If

U, V E V,

U

and

V

denote by

are semilattices

it

follows from Theorem 5 that

U *B V -- U *R V , B

and

R

denoting the varieties of all bands and of regular bands,

respectively.

At the time of writing the

description of the

author knows of no

R-free product in general.

However in

Theorem 4.1 of [2], Olin prescribes

a normal form for the elements

of the free product in the variety

LR

satisfying

the identity

in the variety RR

of left regular bands

ax = axa) and dually for the free product

of right regular bands

(satisfying xa -- axa).

Now it is well-known that on a regular band and

R

relations

L

equivalent

are congruences

to regularity),

S, Green's

(this property in fact being

respectively

the least left regular congruences (disjoint)

on

the least right regular and S.

If

U

regular bands it is then not difficult

u/L ,RR v/L -~ (U *R V)/L are semilattices

then

and dually.

S

L n R

and

V

U

and

V

and dually.

is trivial and

is the spined product

are

to see that

In particular if

U *RR V -~ (U *Rv)/L

On the other hand since any regular band

(those

L v R ~= L~

-- J,

(see [I,V.5]) of S/L

and

S/R associated with the diagram

S

~ S/R

SlL

~ SlJ

.

Combining these results with Theorem 4.1 of [2] and its dual yields a description

THEOREM 6.

If

U

of the B-free product of two semilattices.

and

V

are semilattices

then

U *

- -

V B

isomorphic to the spined product o_~f U *RR V

and

is - -

U *LR V,

their

free products in the varieties of right and of left regular bands respectively.

Every element

b__eeuniquely represented

w

o__ff U *RR V (or of

in the form

340

w = Wl...Wn,

U *LR V)

the terms

can

JONES

belonging alternately to

U

and

V

and satisfying

wi>

wi+2,

1 < i < n - 2.

REFERENCES

i.

Howie, J.M., An Introduction to Semisroup Theory, Academic Press, London, 1976.

2.

Olin, P., Lo$ical Properties of V-free Products of Bands, preprint.

3.

Scheiblich, H.E., The Free Elementary *-orthodox Semigroup, Proc. Monash Semigroup Conference (1979),to appear.

Monash University, Clayton, Victoria, Australia 3168. Received February 14, 1980 and in final form April 7, 1980.

341