Separability properties of semigroups and algebras

Abstract

Separability properties can be seen as generalisations of residual finiteness. In this thesis we investigate four such properties: monogenic subalgebra separability, weak subalgebra separability, strong subalgebra separability and complete separability.

In Chapter 1 we outline the necessary preliminary definitions and results. We define separability properties in terms of universal algebra, in order to be able to study these properties in a range of different settings. We also provide a topological interpretation of these properties. The chapter concludes with the necessary preliminary information to be able to study these properties in semigroups.

In Chapter 2 we investigate the separability properties of free objects in different semigroup varieties. This builds upon work by Hall which shows that the free group is weakly subgroup separable. The varieties considered are groups, semigroups, completely simple semigroups, Clifford semigroups and completely regular semigroups. We also define a new variety, known as α-groups, to aid in our investigation of the free completely simple semigroup.

We begin Chapter 3 by investigating which separability properties are inherited by the Schützenberger groups of a semigroup. We use the theory developed to classify precisely when a finitely generated commutative semi-group has each of four separability properties considered. We conclude the chapter by studying when separability properties of Schützenbeger groups pass to semigroups with finitely many ℋ-classes.

In the final chapter, we consider the preservation of separability properties under various semigroup-theoretic constructions. The constructions considered are the 0-direct union, the direct product, the free product, as well as an investigation into large subsemigroups. We classify precisely when a finite semigroup preserves both monogeinc subsemigroup separability and strong subsemigroup separability in the direct product. We conclude the work by indicating some directions that future research may take.