Finite semigroups with slowly growing p n -sequences

Abstract

The p n -sequence of a semigroup S is said to be polynomially bounded, if there exist a positive constant c and a positive integer r such that the inequality p n (S) ≤ cn r holds for all n ≥ 1. In this paper, we fully describe all finite semigroups having polynomially bounded p n -sequences. First we give a characterization in terms of identities satisfied by these semigroups. In the sequel, this result will allow an insight into the structure of such semigroups. We are going to deal with certain ideals and the construction of ideal extension of semigroups. In addition, we supply an effective procedure for deciding whether a finite semigroup has polynomially bounded p nsequence and give some examples.

Semigroup varieties with a small number of term operations

“…The present paper is a continuation of investigations conducted in [3,6], where finite semigroups (equivalently, finitely generated semigroup varieties) whose p n -sequences are bounded by a polynomial function of n were described. Here we strive for a more general aim of determination of all semigroup varieties (i.e., all semigroups) with such a property.…”

Square extension of a groupoid

“…In [5] all the finite semigroups with bounded p n -sequences are described as nilpotent extensions of semilattices, Boolean groups and rectangular bands, i.e., the ideal extensions of the semigroups by a nilpotent semigroup. All the finite semigroups with polynomially bounded p n -sequences are characterized in [3]. Finally, a necessary and sufficient condition for a variety of semigroups to have a free spectrum with a subexponential rate of growth (to be a log-linear variety) has been found in [4] in terms of identities.…”