On the Berman Conjecture for Finite Semigroups
by
Sinisa Crvenkovic
Institute of Mathematics and Informatics, University of Novi Sad
Coauthors: Igor Dolinka (Institute of Mathematics and Informatics, University of Novi Sad), Nikola Ruskuc (Mathematical Institute, University of St Andrews)

If A is any (finite) algebra, by p_n(A) we denote the number of all n-ary term operations which depend on all of its variables (p₀(A) denotes the number of all constant unary term operations of A). The principal goal of the theory of p_n-sequences is to characterize those sequences of non-negative integers which are representable as the p_n-sequence of some algebra (or, possibly, of an algebra of some specific kind, e.g. of a semigroup). Hence, investigations aiming to establish the way in which the numerical properties of sequences influence the structure of corresponding algebras, have a central importance in this theory.

Towards this distant goal, in 1986 J. Berman formulated an interesting conjecture, claiming that the p_n-sequence of any finite algebra is either bounded above by a constant, or eventually strictly increasing. Ten years later, this conjecture was shown by R. Willard to be false. Still, the Berman Conjecture (BC) holds for a vast number of `natural' algebras, such as monoids, groups, rings, modules, lattices, Boolean algebras, etc. Our interest here is in the restricted BC for semigroups. We present several wide classes of finite semigroups satisfying the property indicated by the BC. For example, these classes include all globally idempotent (S²=S) and all commutative finite semigroups.

Date received: June 29, 2002

Copyright © 2002 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caiv-36.