Combinatorial algorithms in semigroups

Abstract

Computational semigroup theory is concerned with developing and implementing algorithms for determining properties of semigroups. The problems in computational semigroup theory can often be divided into sub-problems in computational group theory and combinatorics. In this thesis, we demonstrate how to split a number of problems into group-theoretical and combinatorial sub-problems, and provide algorithms for solving the combinatorial sub-problems. The algorithms presented in this thesis have been implemented in C++ and GAP, and benchmarks are provided to show them practical.

In Chapter 1, we introduce the necessary background in semigroup theory.

In Chapter 2, we introduce a new algorithm for non-exhaustively determining the structure of a semigroup defined by generators, and show that it applies to certain families of matrices over semirings as well as a number of standard and well-known families to which previous algorithms apply.

In Chapter 3 we describe and compute minimal generating sets for several naturally occurring monoids of boolean matrices, in particular the full boolean matrix monoid, Hall monoid, reflexive boolean matrix monoid, and upper and lower triangular boolean matrix monoids. These results extend the dimensions for which the ranks of these monoids are known. We also determine the rank of the 2 × 2 matrices over the max-plus and min-plus semirings with and without threshold, as well as the n × n matrices over Z/kZ relative to their group of units.

Chapter 4 contains new algorithms for determining the translations and bitranslations of arbitrary finite semigroups. We also provide specialised algorithms for computing translations and bitranslations of semigroups defined by finite presentations, completely 0-simple semigroups, congruence-free semigroups, and completely-simple semigroups.

Finally, Chapter 4 contains some further questions raised by the work in this thesis.