The Berman conjecture is true for nilpotent extensions of regular semigroups

Abstract

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 ES n , while ES 0 is the set of all constant unary term operations of S. Let pn(S) = |ES n | 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 Gratzer and Kisielewicz [7]. The theory of pn-sequence of general algebras was founded by E. Marczewski and his ‘Wroc law 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],