Adventures in applying iteration lemmas

Abstract

The word problem of a finitely generated group is commonly defined to be a formal language over a finite generating set.

The class of finite groups has been characterised as the class of finitely generated groups that have word problem decidable by a finite state automaton.

We give a natural generalisation of the notion of word problem from finitely generated groups to finitely generated semigroups by considering relations of strings. We characterise the class of finite semigroups by the class of finitely generated semigroups whose word problem is decidable by finite state automata.

We then examine the class of semigroups with word problem decidable by asynchronous two tape finite state automata. Algebraic properties of semigroups in this class are considered, towards an algebraic characterisation.

We take the next natural step to further extend the classes of semigroups under consideration to semigroups that have word problem decidable by a finite collection of asynchronous automata working independently.

A central tool used in the derivation of structural results are so-called iteration lemmas.

We define a hierarchy of the considered classes of semigroups and connect our original results with previous research.