  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.
