Automatic semigroup acts

To give a general framework for the theory of automatic groups and semigroups, we introduce the notion of automaticity for semigroup acts. We investigate their basic properties and discuss how the property of being automatic behaves under changing the generators of the acting semigroup and under changing the generators of the semigroup act. In particular, we prove that under some conditions on the acting semigroup, the automaticity of the act is invariant under changing the generators. Since automatic semigroups can be seen as a special case of automatic semigroup acts, our result generalizes and extends the corresponding result on automatic semigroups, where the semigroup S satisfies S=SSS=SS. We also give a geometric approach in terms of the fellow traveller property and discuss the solvability of the equality problem in automatic semigroup acts. Our notion gives rise to a variety of definitions of automaticity depending on the set chosen as a semigroup act and we discuss future research directions

Inverse monoids : decidability and complexity of algebraic questions

This paper investigates the word problem for inverse monoids generated by a set A subject to relations of the form e=f, where e and f are both idempotents in the free inverse monoid generated by A. It is shown that for every fixed monoid of this form the word problem can be solved in polynomial time which solves an open problem of Margolis and Meakin. For the uniform word problem, where the presentation is part of the input, EXPTIME-completeness is shown. For the Cayley-graphs of these monoids, it is shown that the first-order theory with regular path predicates is decidable. Regular path predicates allow to state that there is a path from a node x to a node y that is labeled with a word from some regular language. As a corollary, the decidability of the generalized word problem is deduced. Finally, it is shown that the Cayley-graph of the free inverse monoid has an undecidable monadic second-order theory

Automatic semigroups : constructions and subsemigroups

In this thesis we start by considering conditions under which some standard semigroup constructions preserve automaticity. We first consider Rees matrix semigroups over a semigroup, which we call the base, and work on the following questions: (i) If the base is automatic is the Rees matrix semigroup automatic? (ii) If the Rees matrix semigroup is automatic must the base be automatic as well? We also consider similar questions for Bruck-Reilly extensions of monoids and wreath products of semigroups. Then we consider subsemigroups of free products of semigroups and we study conditions that guarantee them to be automatic. Finally we obtain a description of the subsemigroups of the bicyclic monoid that allow us to study some of their properties, which include finite generation, automaticity and finite presentability