Free Spectra and p_n-sequences of Semigroups
by
Siniša Crvenković
Institute of Mathematics, University of Novi Sad, Novi Sad, Yugoslavia
Coauthors: Igor Dolinka (Institute of Mathematics, University of Novi Sad)

For a semigroup S, p_n(S) denotes the cardinal number of all n-ary operations on S obtained from semigroup words by substitution, which depend on all of its variables. On the other hand, the free spectrum of a semigroup S is the sequence consisting of the sizes of finitely generated (relatively) free semigroups in the variety generated by S. These sequences are not at all independent: they are connected by a simple combinatorial formula. It turns out that these two numerical invariants contain quite a lot of information on the structure of a finite semigroup.

Of course, these concepts can be easily generalized for arbitrary (universal) algebras.

Our main motivating problem is the well-known Berman conjecture from universal algebra, restricted to the semigroup case. It claims that the p_n-sequence of any finite algebra is either bounded above by a constant, or eventually strictly increasing. The general conjecture is proved to be false (by R. Willard in 1996), but it turned out to be true for many natural classes of algebraic systems such as groups, monoids, lattices, rings, idempotent algebras, etc.

We present a number of results confirming the Berman conjecture for certain wide classes of finite semigroups, as well as some related results. In the course of such considerations, p_n-sequences and structural properties of different types of finite semigroups will be brought in a close connection.

Date received: May 5, 2000

Copyright © 2000 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 # caec-16.