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The Berman conjecture is true for nilpotent extensions of regular semigroups. Igor Dolinka and Petar Markovic. Let S be any semigroup. A term operation of S is ...
Algebra univers. 51 (2004) 435–438 0002-5240/04/040435 – 04 DOI 10.1007/s00012-004-1867-z c Birkh¨  auser Verlag, Basel, 2004

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The Berman conjecture is true for nilpotent extensions of regular semigroups ´ Igor Dolinka and Petar Markovic Let S be any semigroup. A term operation of S is an n-ary operation of S which is induced by some nonempty word w by substitution of letters. Such an operation is essentially n-ary if it depends on all of its variables. For n ≥ 1, we denote the set of all essentially n-ary term operations of S by EnS , while E0S is the set of all constant unary term operations of S. Let pn (S) = |EnS | for all n ≥ 0. It is easy to see that the sequence pn (S), n ≥ 0, consists entirely of finite cardinals if and only if S generates a locally finite semigroup variety. Of course, all these concepts can be generalized for arbitrary algebras, cf. the survey paper by Gr¨ atzer and Kisielewicz [7]. The theory of pn -sequence of general algebras was founded by E. Marczewski and his ‘Wroclaw School’ back in the sixties (see [9]). They identified as the main goal of this theory to characterize all sequences of non-negative integers which are representable as the pn -sequence of some algebra (or some particular kind of algebra). Later on, most of the research was limited to finite algebras. One of the most intriguing hypotheses in this direction was given by J. Berman, who conjectured in [1] that the pn -sequence of any finite algebra is either bounded above by a constant, or eventually strictly increasing. While this assertion, referred to as the Berman conjecture, is easily shown to be true for finite monoids, groups, rings, modules, lattices, Boolean algebras, etc., R. Willard refuted this conjecture in [10] by constructing a very nice counterexample, a 4-element algebra with finitely many basic operations whose pn -sequence is (0, 3, 2, 5, 4, 7, 6, 9, 8, . . . ). On the other hand, there are several papers (co-authored by the first named author of this note) investigating the (still open) question whether the Berman conjecture holds in the class of all finite semigroups. Namely, in [5] all finite semigroups whose pn -sequences have constant bounds are determined. In [6], Presented by B. M. Schein. Received January 8, 2004; accepted in final form April 8, 2004. 2000 Mathematics Subject Classification: 08A40, 20M10. Key words and phrases: pn -sequence, Berman conjecture, semigroups. The first named author is supported by Grant No.1227 of the Ministry of Science and Environment of Republic of Serbia. 435

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the Berman conjecture is verified for all finite semigroups whose multiplication is onto (SS = S, usually called globally idempotent semigroups), and for some other particular classes in [4]. The majority of the main results of these papers are formulated in terms of nilpotent ideal extensions of globally idempotent semigroups of some specific kind. This is so because for any finite semigroup S, the sequence S ⊇ S 2 ⊇ · · · ⊇ S m ⊇ S m+1 ⊇ · · · of ideals of S (where S m = {s1 s2 · · · sm : si ∈ S, 1 ≤ i ≤ m}) must eventually stabilize, say at S m = S m+1 . If we denote such S m by S ω , then S ω is globally idempotent, while the Rees quotient S/S ω is nilpotent. The results of [4] cover the cases when S ω is either a union of groups, or left (right) reductive (which, for example, includes all finite commutative semigroups and all finite semigroups in which S ω is inverse). This one-theorem note deals with the case when S ω is regular. Theorem. Let S be a finite semigroup such that S ω is regular. Then either pn (S) is bounded above by a constant, or pn (S) < pn+1 (S) for n large enough. Proof. Let S ω = S m . We prove that the required inequality holds for all n ≥ m (except for the cases described in [5]). In the sequel, we may assume that every term operation under consideration depends on all of its variables. Otherwise, S satisfies a non-regular identity, which by a result of Chrislock [2] (see also Lemma 2.3 of [6]) implies that S ω is a completely simple semigroup. However, in such a case, S ω is a union of groups, and our result follows from Theorem 2.2 of [4]. First, choose an integer r ≥ 1 such that for each a ∈ S belonging to a subgroup of S, ar is the identity element of that group (this can be achieved, for example, by taking r to be the l.c.m. of orders of all subgroups of S). Now we define a mapping S by φ : EnS → En+1 (φf )(x1 , . . . , xn , xn+1 ) = (f (x1 , . . . , xn )xn+1 )r f (x1 , . . . , xn ). By the above remarks, φ is well defined. We proceed by proving that φ is injective. To this end, we shall consider a fixed but arbitrary n-tuple (a1 , . . . , an ) of elements of S. Assume that φf = φg for two (essentially) n-nary term operations f, g of S. For brevity, denote c = f (a1 , . . . , an ), d = g(a1 , . . . , an ). Since n ≥ m, c, d ∈ S ω and thus they are regular. Now, our assumption is that (cx)r c = (dx)r d

(∗)

for all x ∈ S. Let c , d be any inverses for c, d, respectively. By setting x → c and x → d , from (∗) we obtain c = (cc )r c = (dc )r d = d(c d)r

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and d = (dd )r d = (cd )r c = c(d c)r , showing that c H d. Denote by H the H-class of c and d. If H is a group, then (∗) immediately implies c = (cx)r c = (dx)r d = d for any x ∈ H (since cx, dx ∈ H), by the choice of r. So, let H be a non-group H-class. Then choose idempotents e ∈ Rc and h ∈ Lc . It follows that the Hclass Rh ∩ Le contains inverses c, d of c, d, respectively. Since h ∈ Rd ∩ Lc , it follows by Theorem 2.17 of [3] that cd ∈ Rc ∩ Ld = He . Hence, (cd)r = e, and so c = ec = (cd)r c = (dd)r d = d, by (∗). Since the parameters (a1 , . . . , an ) were chosen arbitrarily, we conclude that f = g. Therefore, it remains to exhibit a term operation of S which is not of the form φf for any f ∈ EnS . We claim that the operation x1 · · · xn xn+1 will do. We may take for granted that S has a non-group H-class H, for otherwise Theorem 2.2 of [4] provides the required result, as before. Let a ∈ H be arbitrary, let e ∈ Ra be an idempotent, and set xn+1 → a, while all other variables are evaluated to e. The assumption x1 · · · xn xn+1 = φf yields a = ea = en a = (f (e, . . . , e)a)r f (e, . . . , e) = (ea)r e = ar e = (ae)r . However, since Ha = Re ∩ La contains no idempotent, by [3, Theorem 2.17] we have that ae ∈ Ra ∩ Le (= He ). On the other hand, Rae ≤ Ra and Lae ≤ Le , and the preceding argument shows that at least one of these inequalities must be strict. Since the R-classes (L-classes) contained in the same D-class of a finite semigroup must be incomparable (cf. Proposition 3.7 of [8]), ae ∈ Da , whence a = (ae)r ∈ Da . The contradiction just obtained completes the proof of the theorem.  References [1] J. Berman, Free Spectra of Finite Algebras, Lecture Notes, University of Illinois at Chicago, 1986. [2] J. L. Chrislock, A certain class of identities on semigroups, Proc. Amer. Math. Soc. 21 (1969), 189–190. [3] A. H. Clifford and G. P. Preston, The Algebraic Theory of Semigroups, Vol. I, Amer. Math. Soc., Providence, 1961. [4] S. Crvenkovi´c, I. Dolinka and Ruˇskuc, N., Notes on the number of operations of finite semigroups, Acta Sci. Math. (Szeged) 66 (2000), 23–31. [5] S. Crvenkovi´c, I. Dolinka and Ruˇskuc, N., Finite semigroups with few term operations, J. Pure Appl. Algebra 157 (2001), 205–214. [6] S. Crvenkovi´c, I. Dolinka and N. Ruˇskuc, The Berman conjecture is true for finite surjective semigroups and their inflations, Semigroup Forum 62 (2001), 103–114. [7] G. Gr¨ atzer and A. Kisielewicz, A survey of some open problems on pn -sequences and free spectra of algebras and varieties, in: Universal Algebra and Quasigroup Theory, A. Romanowska and J. D. H. Smith eds., Heldermann Verlag, Berlin, 1992, pp. 57–88. [8] G. Lallement, Semigroups and Combinatorial Applications, Wiley, New York, 1979.

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[9] E. Marczewski, Independence in abstract algebras. Results and problems, Colloq. Math. 14 (1966), 169–188. [10] R. Willard, Essential arities of term operations in finite algebras, Discrete Math. 149 (1996), 239–259. ´ Igor Dolinka and Petar Markovic Department of Mathematics and Informatics, University of Novi Sad, Trg Dositeja Obradovi´ca 4, 21000 Novi Sad, Serbia and Montenegro e-mail : [email protected], [email protected]

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