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/ . Austral. Math. Soc. {Series A) 51 (1991), 381-399

TOTALLY COMMUTATIVE SEMIGROUPS

J6ZEF DUDEK and ANDRZEJ KISIELEWICZ

(Received 4 April 1989; revised 23 February 1990)

Communicated by P. G. Trotter

Abstract

A semigroup is totally commutative if each of its essentially binary polynomials is commutative, or equivalently, if in every polynomial (word) every two essential variables commute. In the present paper we describe all varieties (equational classes) of totally commutative semigroups, lattices of subvarieties for any variety, and their free spectra.

1991 Mathematics subject classification (Amer. Math. Soc.) 20 M 07, 20 M 05, 08 B 15; Sec- ondary 20 M 14, 08 B 05.

In [2, 7, 8] all varieties of idempotent semigroups are described. A semigroup (S, xy) is idempotent, if xy is idempotent, that is the identity x2 = x holds. Any idempotent semigroup is totally idempotent in the sense that each of its binary polynomials is idempotent, as well. In this paper we consider from this point of view the commutative law

(1) Ax, y) = f(y, x).

In contrast with idempotency, this identity is interesting only for operations depending on both the variables (that is, essentially binary), since otherwise (1) fails to hold, unless f(x, y) is a constant operation.

DEFINITION. A semigroup S is totally commutative (in short, a TC-semi- group) if for every essentially binary polynomial f(x, y) of S the identity (1) holds.

Boolean groups and semilattices are important and well-known examples of TC-semigroups. Left and right zero semigroups are examples of improper TC-semigroups (see [17] and Section 2 below).

© 1991 Australian Mathematical Society 0263-6115/91 $A2.00 + 0.00

381

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382 Jozef Dudek and Andrzej Kisielewicz [2]

We show that the class of TC-semigroups is the union of four equational classes. There are No distinct varieties of TC-semigroups and all the varieties are finitely based. For each variety of TC-semigroups normal forms for words are found, solutions for word problems are explicitly shown and the lattices of subvarieties are completely described. For these kind of problems considered for various classes of semigroups see [10, 11, 12, 15] and [19] for further references. As in [9], we also describe pn -sequences of all the varieties, and in consequence, cardinalities of free semigroups (free spectra). From this point of view the class of TC-semigroups is especially interesting, providing the most natural examples of free spectra with the slowest growth (for this topic see also [5, 13, 16]). Finally, in the last section, a certain contribution to the topic of hyperidentities is given (see [1, 3, 4, 18]).

For the terminology used in this paper the reader is referred to [19].

1. General remarks

A variable xt in a polynomial p — p{xx, ... , xn) is called essential if the value of p does really depend on xt; for a more precise definition see [13]).

PROPOSITION 1.1. 4̂ semigroup S is a TC-semigroup if and only if every polynomial p = p{xx, ... , xn) over S is symmetric in all essential variables.

PROOF. The " i f part is trivial. For the "only i f part let S=(X, xy) and note that either xy is not essentially binary (then each polynomial in S de- pends on at most one variable and the result is trivial) or xy depends on both the variables, and in consequence S is commutative. Let p = xf'x"2 • • -x°", be an arbitrary polynomial of S and assume that p depends on each of the variables xt. It follows that xf'x"

J is an essentially binary operation, and by the assumption, x?'x°J = xfx? . Hence p = x^x^ •••xan"n for any permutation n of indices 1, 2 , . . . , « , as required.

We note that the equivalence in Proposition 1.1 does not hold for nonas- sociative groupoids. A counterexample is the class of groupoids considered in [14].

PROPOSITION 1.2. A semigroup S = (S, xy) is a TC-semigroup if and only if one of the following conditions holds identically:

(i) xy = x2; (ii) xy = y2; (iii) xy = yx and xy2 = x3; (iv) xy = yx and xy = x y.

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[3] Totally commutative semigroups 383

PROOF. We prove the "only i f part. At first, note that if xy is not es- sentially binary, then either xy = xz or xy = zy, and consequently the condition (i) or (ii) holds. If xy is essentially binary, then by definition xy = yx. Now, consider the polynomial xy . If it is also essentially binary, then xy2 = x2y, and the condition (iv) holds. If xy2 is not essentially bi- nary, then either xy2 = xz2 or xy2 = zy2 holds. In the former, xy2 = x3, so (iii) holds. In the latter xy2 = y3 and consequently x2y2 = y3 and, by commutativity, y3 = x3. Hence, again xy2 = x3, which proves the "only i f part.

To prove the " i f part, note at first, that if the condition (i) or (ii) is sat- isfied, then there is no essentially binary polynomial in S, and consequently S is totally commutative ((i) and (ii) are thus trivial cases).

Assume that the condition (iv) holds. Then, it is easy to deduce by simple induction, that xkym = xk~xym+l for any k > m > 0, and in consequence xkym = xmyk . In particular, S is a TC-semigroup.

It remains to consider the condition (iii). First we note that xy = x3 for any k > 0. Consequently, xmy2k and x2kym are not essentially binary. On the other hand, substituting xy for x in xy2 = x3, we obtain xy3 = {xy)3 = x3y, and consequently, as above, x ym = xmyk for any odd k > m > 0. Hence, every polynomial xkym either does not depend on one of the variables or is symmetric, thus completing the proof.

2. Improper semigroups

DEFINITION. A semigroup S = (S, xy) is proper if xy is essentially bi- nary. Otherwise, it is called improper.

By the proof of Proposition 1.2, S is improper if and only if the con- dition (i) or (ii) holds. In particular, every improper semigroup is totally commutative. It is an easy exercise to find normal forms for words in im- proper semigroups and describe the lattices of subvarieties. Figure 1 shows both the lattices. The subvarieties given by xy — x , xy = z2 , xy - y are known respectively as left zero semigroups, constant semigroups, right zero semigroups, and trivial semigroups are at the bottom (see [17]).

PROPOSITION 2.1. Let S be an improper semigroup. Denote (p(x) = x2

and T = of S onto T satisfying q>{x) = JC for x e T, define xy =

384 Jozef Dudek and Andrzej Kisielewicz

xy = x2

xy = z-

[4]

Moreover, S is a left (or right) zero semigroup if and only if T = S; S is a constant semigroup if and only if the cardinality \T\ = 1; and S is trivial if and only if \S\ = l.

PROOF. This is trivial to prove.

3. TC-semigroups vrithxy2 = x3

In this section we consider the variety G of semigroups denned by the laws xy = yx and xy2 — x3 (the case (iii) in Proposition 1.2). All deductions are carried out in Eq G (the equational theory of G). In particular, equivalent identities means here equivalent in Eq G. Also x ,y, z stand for variables, while a, x, 6 stand always for words (terms of semigroups).

At first, we observe that the word x6 induces a constant polynomial in every member of G. Indeed, from xy2 = x3 we have x2y2 = x6 and by commutativity x2 = y2x2 = y6, as required. Hence, in the sequel, we will write C for x6.

In turn, we establish some simple laws holding in G. We have 2k

2k+\ = X

3 for every k > 1,

(i) xLK = C for every k > 1,

(ii) (iii) (iv)

Indeed, from

x 3y3 = x3y = y3x,

= x2y2 we have x6 = x4 and (i) follows by simple induction. From (i) we have x

x3 2k+l = Cx for k > 1 which combined with

x5 = x(x2)2 = yields (ii) and (iii). Finally, x3y3 = x2y(xy2) = x5y combined with (ii) yields (iv).

Now, we find normal forms for words in G.

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[5] Totally commutative semigroups 385

LEMMA 3.1. For every a , , . . . , an > 1, a = x" 1 • • • x°n is equal to one and

only one of the following words: (a) x { • • • x n whenever ax = • • • = an = 1 ; (b) C whenever all a , , . . . , an are even and either n > 1 or a{ > 2; (c) jcf whenever n = 1 and al = 2; (d) x* xt • • • xt when a, , . . . , a, are all odd numbers among a , , . . . , an

( / , < • • • < ik) and k>\.

PROOF. We prove that a is equal to one of words given in (a) to (d). At first, note that by (i), C2 = C. Now, using (i) it is easy to see that (b) holds. Also (a) and (c) are trivial. In the remaining case, by (b), a = CX:' • • • x> , where (b,, . . . , bt) = (a,