A basis for the type

Aequationes Mathematicae 49 (1995) 205-213 University of Waterloo

0001-9054/95/030205-09 $1.50 + 0.20/0 © 1995 Birkh/iuser Verlag, Basel

A basis for the type ( n ) hyperidentities of the medial variety of semigroups SHELLY L. WISMATH* Summary. The medical variety M V of semigroups is the variety defined by the medial identity xyzw = xzyw. This variety is known to satisfy the medial hyperidentities F(G(xll . . . . . x~n) . . . . . G(x,l . . . . . xnn)) = G(F(xII . . . . . Xnl). . . . . F(x 1. . . . . .

x~)),

for n -> 1. Taylor has observed in [2] that M V also satisfies some other hyperidentities, which are not consequences of the medial ones. In [4] the author introduced a countably infinite family of binary hyperidentities called transposition hyperidentities, which are natural generalizations of the n = 2 medial hyperidentity. It was shown that this family is irredundant, and that no finite basis is possible for the M V hyperidentities with one binary operation symbol. In this paper, we generalize the concept of a transposition hyperidentity, and extend it to cover arbitrary arity n -> 2. We show that the M V hyperidentities with one n-ary operation symbol have no finite basis, but do have a countably infinite basis consisting of these transposition hyperidentities.

1. I n t r o d u c t i o n

A h y p e r i d e n t i t y o f type z is f o r m a l l y the s a m e as a n i d e n t i t y o f that type. A n i d e n t i t y P = Q is said to be hypersatisfied by a variety V, n o t necessarily o f the s a m e type if, w h e n e v e r the o p e r a t i o n s y m b o l s o f P a n d Q are replaced b y the terms o f V o f the a p p r o p r i a t e arity, the i d e n t i t y w h i c h results holds in V. F o r e x a m p l e , the type ( 2 ) i d e n t i t y F ( x , x ) = x is hypersatisfied by a n y variety all o f w h o s e b i n a r y terms are i d e m p o t e n t . W h e n V hypersatisfies P = Q, we refer to P a n d Q as h y p e r t e r m s , a n d to P = Q as a h y p e r i d e n t i t y satisfied b y V. T h e reader is referred to W. T a y l o r ([2]) for a m o r e c o m p l e t e i n t r o d u c t i o n to hyperidentities.

* Research supported by NSERC of Canada. AMS 1991 subject classification: 08, 20M. Manuscript receiued September 3, 1992 and, in final form, August 14, 1994.

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In this paper we look at the type ( n ) hyperidentities (hyperidentities with one operation symbol of arity n) satisfied by the medial variety of semigroups. The medial variety M V is defined as the variety of semigroups satisfying the medial identity x y z w = x z y w . This variety is particularly significant as the hyperidentity closure of the variety A of commutative semigroups, in the sense defined in [3]: A and M V satisfy precisely the same hyperidentities. In fact, the uncountably infinite number of varieties of semigroups between A and M V all share this property. The variety M V is known to satisfy the corresponding type (2) medial hyperidentity, F ( F ( x , y), F(z, w)) = F ( F ( x , z), F ( y , w)),

and more generally the type ( n ) version F(F(xll .....

xl.) .....

= F(F(Xll . . . . .

F(x~l . . . . .

x.l) .....

x..))

F(Xl . . . . . .

Xnn)),

for n > 2.

However, as Taylor observed in [2], M V also satisfies other hyperidentities which are not consequences of these medial ones. In [4], the author introduced a countably infinite family of type (2) hyperidentities called transposition hyperidentities, which are in some sense the natural generalization of the type (2) medial hyperidentity. We showed that this family was irredundant, and could not be deduced from any finite set of type (2) hyperidentities. This means that no finite basis is possible for the M V hyperidentities of type (2). The goal of this paper is to generalize and extend this result. For each type (n), n -> 2, we define a family of hyperidentities called transposition hyperidentities. These are particularly simple in structure, and are natural generalizations of the basic type ( n ) medial hyperidentity. We first prove that for each n >-2, the collection ~ , of all type ( n ) transposition hyperidentities forms a countably infinite basis for the type ( n ) hyperidentities of M V . Then using a suitably chosen subset of ~ , we prove that no finite basis is possible. In Section 2, we introduce the transposition hyperidentities, and state the results which lead to the main theorem. Of particular importance is a characterization by Taylor of which hyperidentities are satisfied by M V . The proofs themselves are rather technical, and are presented in Section 3.

2. The transposition hyperidentities The main tool to be used in analyzing M V hyperidentities is a characterization by Taylor of precisely what hyperidentities M V satisfies. Let P be any hyperterm of

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type ( n ) . P can be represented by a formation tree diagram, where each leaf node is labelled by a variable, and each non-leaf node corresponds to the n-ary operation symbol (which by convention is omitted from the diagram). For any leaf node in P, we can encode a description of the path from the root of the tree to that leaf node: we use a sequence 7= ill2"" ik, k-> 1, where ij e {1, 2 . . . . . n}, and the j t h entry in the sequence represents the branch taken at the j t h stage in the path. Such a sequence will be called a path or a path description in P, of length k. We may as convenient also indicate at the end of the path the variable occurring on the leaf node, as for instance ? = i~i2"''ikX. The complete list of all such path descriptions, for all leaf nodes in P, will be called the path list for P. For example, the hyperterm y p=

x

has path list

z s

lx 21y

t W

22z 23r 31s 32t 33w.

Now let P = Q be any type ( n ) hyperidentity. We will say that P = Q is

balanced if there is a bijection tr from the path list of P to the path list of Q, with the property that, for any path description d, ~-(d) is just a rearrangement of d. LEMMA 1. (Taylor, [2]) M V satisfies a type (n ) hyperidentity iff the hyperiden-

tity is balanced. We now focus on this balanced property to define a particular kind of hyperidentity. Let n be > 2, and let 7 = i~/2 • • • ik be any finite sequence of digits from {1, 2 . . . . , n}. Let j = j l j 2 ' " "Jk be any rearrangement of ~. Let P be the smallest possible type ( n ) hyperterm having path descriptions il i2"'" ika and JlJ2"" "jkb in its path lists. Let Q be the hyperterm obtained from P by swapping or transposing the two variables a and b, and leaving everything else alone. We will call the hyperidentity P = Q the transposition hyperidentity induced by 7 and ], and label it

T(n; ~,]). To illustrate this construction, let n = 3 , 7 = 1231 and ] = 3 2 1 1 .

Then the

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induced hyperidentity is a

b

b

a

(Note that the unlabelled leaf nodes here are assumed to carry the same label in both hyperterms.) It is clear that such transposition hyperidentities are balanced, since the path lists on either side are identical except for a swap of descriptions

~i, i2""ika [J,Jz" "jkb

~i, iz"'il, b [J,Jz "jka

with

Thus our transposition hyperidentities are all satisfied by MV. Note that in the transposition notation, the basic type (2> medial hyperidentity is just T(2; 12, 21); we are in a sense now generalizing this basic transposition by allowing swaps defined by sequences other than 12 and 21. Moreover, the general type (n> medial hyperidentity for n > 3, although not a transposition, can be obtained as a composition of transpositions. For instance, for n = 3, u

v

w

y

v

b W

is easily seen to be a composition of the transpositions T(3; 12, 21), T(3; 13, 31) and T(3; 23, 32). Notice that hyperidentities like this one which are compositions of one or more transpositions have the property that the two hyperterms have the same shape. However, hyperidentities which are consequences (according to the equational logic for hyperidentities, as described by Graczynska and Schweigert in [1]) of the transpositions need not always have this property. An an example of this, consider the hyperidentity

Y

x

b

c

a

y

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It is a consequence of the transposition T(2; 21, 12), since applying the 21, 12 swap to the path list 1Ix

leads to path list

1lx ~121b [ 122c

12a

211b 212cj

21a

22y

22y.

Here the one path 12a is swapped with the whole 21-branch, that is, with the longer paths 2 l i b and 212c. We define ~ , to be the collection of all possible type (n> transposition hyperidentities. We will show that ~ , gives a basis for the M V type (n > hyperidentities. ~ , is not irredundant; for example, we can deduce T(2; 122, 212) from T(2; 12, 21) and T(2; 121, 211). However, we will show that no finite basis for M V type ( n ) hyperidentities is possible. For this, we will use a particular subset of ~ , . First, let us define P,k to be the smallest hyperterm of type (n > containing paths 1---1 ~ n .. v. ._n_a,n d n . .~__.v.__, .n 1 . - . 1 . We define Q,k to be the same hyperterm as P, k ~__v__.J ~___.v__~ k

k

k

k

except that the variables on the two special paths are swapped. That is, P~k = Q~k is the transposition T(n; ~ . . ~ n__..H~ , n . . . n 1 - . . 1), k > 1. Now let us define k

k

F, to be the collection of all the hyperidentities P,k = Q,k, for k > 1. LEMMA 2. I f M V satifises a type (n> hyperidentity P = Q, then there is a finite sequence o f transpositions leading f r o m P to Q. LEMMA 3. Let P,k and Q~k be the hyperterms above, f o r n > 2 and k > 1. I f M V satisfies P~k = R, f o r some type ( n ) hyperterm R, then R must be one of: (i) e n k (ii) Qnk : the hyperterm obtained f r o m P,k by transposing the 1 • • • In • • • n with the n . • • n 1. • • 1 variable. (iii) the hyperterm obtained f r o m P,k by transposing the In with the n 1 variable. or (iv) the hyperterm obtained f r o m Pnk by transposing the 1 . . . In . . . n with the n • • • n 1 • •. 1 and the In with the n 1 variable.

We will illustrate Lemma 3 with an example. Taking n = 3 and k = 3, we have

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P,k equal to

l ~ ~ J j

Z W ~

L e m m a 3 says that if M V is to satisfy a hyperidentity P.k = R, then we can form R from P in exactly four ways: do nothing to P; transpose variables z and w in P; transpose variables x and y in P; or transpose both z with w and x with y. COROLLARY 4. The collection Fn cannot be deduced from any finite set of

hyperiden titles. These results combine to give us THEOREM 5. The type ( n ) hyperidentities of the medial variety M V have a countably infinite basis ~n, and no finite basis is possible.

3. Proofs of the Lemmas In this section we prove Lemmas 2 and 3, as described above.

Proof of Lemma 2. Let P = Q be a type ( n ) hyperidentity satisfied by MV. By L e m m a 1, P = Q is balanced, and there is a bijection ~ from the path list of P to that of Q. We will use a to produce a finite sequence, P = P0, P1 . . . . , P,~ = Q of hyperterms, such that each P t + l is obtained from Pt by making one transposition of variables (0 < t < m - 1). Our method is to use at each stage t a transposition T(n; ~, o-(0), where ~ is the shortest path sequence in Pt_ 1 such that o ( ~ ) ~ ~. (Informally, we work to transform the path list for P into that for Q, one transposition at a time, by always doing the shortest possible swap available.) If more than one path sequence of minimal length needs to be swapped, we choose any one of that length.

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To prove that this method always works to transform P into Q using only transpositions, we need to verify one key thing. Suppose that at some stage t, 1 < t < m, we cempare P,_ ~ with Q, and find that the shortest path to be swapped is i = i l i 2 " ' ' i k a in P,_ 1 w i t h j = j l . . "jka in Q. In order to use the transposition T(n; ~,]) to form Pt f r o m P t - 1 , we m u s t ensure the existence of a path jl " " ] k - - i n P t - 1. Here the -- is meant to indicate that this path could have length > k; that is, our swap at stage t could swap the path ~ with a path of the same or longer length. (We have seen in an example in the previous section how this can happen.) In particular, this guarantees that our process has no loops in it: any swap done at stage t will alter paths of length > k, but will not damage any shorter paths already dealt with at earlier stages. We now give a p r o o f of the existence of the required path. We assume that at some stage t, the path ~ = il " • ' ik in P,_ 1 is paired by ¢r w i t h ] =Jl ' " " J k in Q, and show that P t - 1 must have a path of length > k, beginning with j~ • • • Jk. Suppose that ~ and ] agree in the first p places with 0 < p - k - 1, differing for the first time in the (p + 1)st place. Then in P,_ ~ there must be a path labelled J l ' " .]pip+ ~ - - b and hence one labelled Jl"" " j p j p + l - - b , where - - i n d i c a t e s a (possibly empty) sequence from {1, 2 . . . . . n } and b is a variable. If p = k - 1, we have the required path. If not, we examine j~. • .jpjp + ~ - - b . If we had just j ~ . • .jpjp + 1b, with - - the empty sequence, this path in P , _ ~ must map to a non-trivial rearrangement of itself, or we would have dealt with it at an earlier stage. However, if it is paired to itself in Q, this would mean that J 1 " ' "Jp+lb represents a "dead-end" in Q, contradicting our assumption that Q contains path J = J l ' " " J ~ J p + l ' ' ' J k with p < k - 1. This shows that the - - cannot be empty, s o P t - l must contain Jl " " J p J p + l l - - b , for any l ~ {1, 2 , . . . , n }, including l =jp + 2, and again with - possibly empty. I f p + 2 = k, we are done; if not, we repeat the above argument to show that this new - - cannot be empty. Continuing in this way we get a path Jl • " "JpJp + 1 " " " A - - in Pt - 1, as required. [] P r o o f o f L e m m a 3. Let M V satisfy a hyperidentity P,k = R, for some type (n> hyperterm R. By L e m m a 1, there is a bijection cr from the path list of P,k to that of R, mapping each path description ~ to a rearrangement ~(~). We will consider in turn each of the various possible path descriptions ~ from Pnk, and show that except in two cases, we must have a(~) = 7. These two exceptional cases will lead to the four possibilities for the formation of R, as described in the statement of the Lemma. Our consideration of the paths in P.k will proceed by increasing path length, starting with paths of length one. Clearly, for ~ = j , j ~ {2, 3 . . . . . n - 1}, we must have a(?) = i. That is, in R there are the same length one paths 2, 3 . . . . . n - 1. Moreover, this guarantees that the path descriptions 2, 3 . . . . , n - 1 are "dead

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e n d s " in R : n o l o n g e r p a t h d e s c r i p t i o n s c a n s t a r t w i t h a n y o f these n u m b e r s . T h e s e d e a d e n d s t h e n e n s u r e t h a t for l e n g t h t w o p a t h s o f the f o r m ~ = l j or ~ = n j , j E {2, 3 , . . . , n - 1}, we m u s t also h a v e ~'(~) = ?, a n d d e a d e n d s o c c u r here too. T h e r e are t w o o t h e r l e n g t h t w o p a t h s : for ~ = In a n d ] = n 1, we can h a v e e i t h e r a ( 0 = ~ a n d ~ ( j - ) = L o r a ( ~ ) = f a n d a ( ] ) = ?. T h e s e t w o cases a m o u n t to the possibilities o f leaving Pnk u n c h a n g e d , o r t r a n s p o s i n g the In a n d n 1 v a r i a b l e s . In either case, the In a n d n I p a t h s m u s t also be d e a d e n d s in R. A t l e n g t h three, we have P,k p a t h s o f the f o r m ~ = l l j o r ~ = n n l , j e { 2 , 3 . . . . , n } a n d l ~ {1, 2 . . . . . n - 1}. T h a t these are all d e a d e n d s follows f r o m the p r e v i o u s l e n g t h one a n d t w o d e a d ends. In general, it follows t h a t f o r ~ = 1 . - . l j or [ = n " . . n l , w i t h t l's, w h e r e j e {2, 3 . . . . . n}, l ~ {1, 2 . . . . . n - 1}, a n d 1 < t < k - l, we h a v e a d e a d e n d at ~in R; in all cases e x c e p t the l n / n l s w a p m e n t i o n e d a b o v e , a(D = 7. T h i s c o m p l e t e s the c o n s i d e r a t i o n o f p a t h s o f l e n g t h < k in P,k. A t l e n g t h k + l we s t a r t w i t h the t w o p a t h s 1 • • • l a n d n n • • • n , b o t h clearly h a v i n g no n o n t r i v i a l r e a r r a n g e m e n t s , a n d t h u s giving d e a d e n d s in R. The o t h e r l e n g t h k + l p a t h s h a v e the f o r m ~ = 1 • • • l j o r d u a l l y ~ = n • • • n j , with j ~ {2, 3 . . . . . n - 1 }. A g a i n , the p r e v i o u s d e a d e n d s rule o u t r e a r r a n g e m e n t s j l . . . 1, l j l . . . 1, 1 l j l . . . 1, a n d so on, f o r c i n g a(t-) = ? a n d a d e a d end a t ~ in R. At length k+2, we h a v e paths o f the f o r m ?=l.-.lnj, with j e {1, 2 . . . . . n - 1}, a n d d u a l l y ~ = n n . . • n l l , w i t h I e {2, 3, . . . , n}. W e will s h o w t h a t for ? = 1 • • • l n j , the o n l y r e a r r a n g e m e n t a(~) o f ~ p o s s i b l e is g itself; the d u a l s i t u a t i o n is h a n d l e d similarly. F i r s t , s u p p o s e j = 1. F r o m a b o v e , we h a v e r u l e d o u t n l , ln, l l n . . . . . 1 - . - l n l l and 1.-.In as d e a d ends, so we m u s t have a f t ) = 1 . - - l n l = g. N o w s u p p o s e j e {2, . . . , n - 1}. Since j, n l a n d n j are all d e a d ends, we see t h a t a(t-) m u s t h a v e a 1 in its first p o s i t i o n . T h e n In a n d l j b o t h d e a d e n d s forces tr(t-) to s t a r t with 11. G e n e r a l l y , we h a v e d e a d e n d s a t 1 • . . l j a n d 1 .. • In for t l's, 1 < t < k - 1, so the first k entries in a(~) m u s t b e 1. This forces a ( O to be e i t h e r ? itself o r 1 • • • l j n . Since 1 • • • l j ( w i t h k l ' s ) is a d e a d end f r o m the length k + 1 stage, we c a n e l i m i n a t e this p o s s i b i l i t y . T h u s we m u s t a g a i n h a v e a(~) = ~. T h i s a r g u m e n t n o w generalizes t o s h o w t h a t aft) = ~ for a n y ~ o f the f o r m ~ w h e r e j e { 1, 2 . . . . , n - 1 } a n d 1 < t < k - 1. F o r e a c h t - v a l u e , we •

t

use all the d e a d e n d s o b t a i n e d a t p r e v i o u s stages to e l i m i n a t e all possibilities for a ( 0 e x c e p t i itself. F i n a l l y , the last p a t h to c o n s i d e r is ~ = 1 • • • In • • • n, o f l e n g t h 2k. W e h a v e e a r l i e r d e a d e n d s at ln, 11n . . . . . 1 • • • 1 n , so if a ( 0 is to have a i as its first entry, we m u s t h a v e ~r(~) = 1 • • • In • • • n = ~. D u a l l y , if a ( ~ h a s n as its first e n t r y , d e a d e n d s a t n I, n n 1 . . . . . • • • 1 e l i m i n a t e all b u t a f t ) = n • • • n 1 -. • 1.

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W e have n o w shown that, for almost all path sequences ~ in Pnk, we m u s t have a(i) = 7. There are two exceptions possible: a ( l n ) could be either In or n l, and tr1 ~ - ~ nf _ . ~ ) could be either 1 . . . In . . . n or n . . - n l . - . 1. k

k

This gives us exactly four possible ways to define the bijection a. F r o m this we see that there are four possible hyperterms R such that M V will satisfy P,,k = R. [] REFERENCES

[l] GRACZYNSKA, E. and D. SCHWEIGERT, Hyperidentities of a given type. Algebra Universa3is 27 (1990), 305-318. [2] TAYLOR, W., Hyperidentities and hypervarieties. Aequationes Math. 23 (1983), 30-49. [3] WISMATH, S. L., Hyperidentities for some varieties of commutative semigroups. Algebra Universalis 28 (1991), 245-273. [4] WlSMATH, S. L., On finite hyperidentity bases for varieties of semigroups. To appear in Algebra Universalis.

Department of Mathematics & Computer Science, University of Lethbridge, Lethbridge, Alberta, Canada T1K 3M4.