Semigroup and Category-Theoretic Approaches to Partial Symmetry

TL;DR

This thesis explores partial symmetry through semigroup and category theory, establishing new connections between self-similar group actions, HNN-extensions, and operator K-theory, with implications for fractals and algebraic structures.

Contribution

It introduces novel categorical frameworks linking partial symmetries, self-similar actions, and HNN-extensions, and develops a functor connecting inverse semigroups with abelian groups.

Findings

Rees monoids are monoid HNN-extensions.

Self-similar group actions generate specific monoids from fractals.

The K-functor relates inverse semigroup categories to abelian groups, matching known K-theory groups.

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.