A note on automatic semigroups

Abstract

It is well known that for monoids and completely simple semigroups the property of being automatic does not depend on the choice of the semigroup generating set. In this paper we extend these results to semigroups satisfying the equality S=SS. As an immediate corollary of this result we obtain that for regular semigroups being automatic is invariant under the change of generators

Automatic semigroup acts

“…In the sequel we aim to strengthen the most general result currently existing, namely that of [8]. The main idea is to show that under a certain condition, almost all words over one generators can be rewritten to words over the other generators in both a short and a long way, which enables one to use Lemma 3.9.…”

“…For example, in [5] it is proved that automatic completely simple semigroups are characterized by the fellow traveller property, that they are finitely presented and that automaticity does not depend on the choice of the generating set. In [8], it is proved that for semigroups satisfying that S = SS, the property of being automatic does not depend on the choice of the semigroup generating set. On the other hand, the research community experimented with alternative definitions [12,18,17].…”

“…On one hand semigroup classes related to groups have been investigated [1,2,5,7,10,8]. For example, in [5] it is proved that automatic completely simple semigroups are characterized by the fellow traveller property, that they are finitely presented and that automaticity does not depend on the choice of the generating set.…”

“…The behaviour of automaticity with respect to changing the generators has been intensively investigated. It was shown that automaticity is invariant under changing the generators for groups [11], for monoids [10], for semigroups with local right identities [9] and for semigroups satisfying S = S 2 [8]. In case of S-acts, there are two types of generators: one for the S-act and one for S. We show that automaticity is invariant under changing the generators of the S-act and we show that if S is a semigroup satisfying that S N = S N +1 for some N ∈ N, then automaticity is invariant under changing the generators of S. Next, we discuss the equality problem for automatic S-acts.…”