Semigroup and Category-Theoretic Approaches to Partial Symmetry

Abstract: This thesis is about trying to understand various aspects of partial symmetry using ideas from semigroup and category theory. In Chapter 2 it is shown that the left Rees monoids underlying self-similar group actions are precisely monoid HNN-extensions. In particular it is shown that every group HNN-extension arises from a self-similar group action. Examples of these monoids are constructed from fractals. These ideas are generalised in Chapter 3 to a correspondence between left Rees categories, self-similar groupoid actions and category HNN-extensions of groupoids, leading to a deeper relationship with Bass-Serre theory. In Chapter 4 of this thesis a functor $K$ between the category of orthogonally complete inverse semigroups and the category of abelian groups is constructed in two ways, one in terms of idempotent matrices and the other in terms of modules over inverse semigroups, and these are shown to be equivalent. It is found that the $K$-group of a Cuntz-Krieger semigroup of a directed graph $G$ is isomorphic to the operator $K^{0}$-group of the Cuntz-Krieger algebra of $G$ and the $K$-group of a Boolean algebra is isomorphic to the topological $K^{0}$-group of the corresponding Boolean space under Stone duality.