Modern computational methods for finitely presented monoids

Abstract

In this thesis we are mainly interested in the development of practical algorithms for semigroups and monoids defined by finite presentations. Although in general nearly every problem about finitely presented semigroups is undecidable, many finitely presented semigroups and monoids of interest are more tractable. Semigroup and monoid presentations have been widely studied in the literature more or less since the inception of the field of semigroup theory. The aim of computational semigroup theory, of which this thesis forms a part, is to develop algorithms and software tools for computing with semigroups, and on the applications of these tools to research problems.

In this thesis we develop the concept of words graphs, which form the basis for the work presented in the first half of the thesis. We describe an algorithm that computes one-sided congruences of finitely presented semigroups. This is the semigroup theoretic analogue of an algorithm described by Sims for computing subgroups of small index in finitely presented groups. Furthermore, we focus on the Todd-Coxeter Algorithm, one of the most widely studied algorithms in computational semigroup theory. We describe a more general version of the Todd-Coxeter Algorithm than the versions available in the literature for computing congruences of finitely presented semigroups.

The remaining part of this thesis is focused on a class of finitely presented monoids, called small overlap monoids. These are, in some sense, the generic finitely presented monoids. They have decidable word problem that can be solved in linear time. We present the results related to the word problem and the combinatorial theory for small overlap monoids developed by Kambites. In addition, we discuss methods appearing in the literature for normal forms in small overlap monoids and we present a new practical algorithm for computing normal forms.