### Abstract

Semigroup theory, and subsequently inverse semigroup theory, is a broad field that is part of many different areas due to its generality. For example, it can be found in parts of computer science, more specifically the concepts of finite state automata and the study of formal languages.

This paper shall discuss the history of the field, and then we shall
discuss some basic definitions of groups and semigroups to lay the foundations needed for
the rest of this report. From there we will discuss the partial order, which is a concept that
arises often in regards to semigroups and inverse semigroups. Then we shall discuss some
well-known theorems, namely Cayley’s theorem, an important theorem for groups, and the
semigroup analogue in the faithful representation theorem for full transformation monoids.

After this we shall discuss inverse semigroups, explore the natural partial order, which is a
partial order restricted to *E(S)* which works very nicely in inverse semigroups. Then this
will lead us to discussing symmetric inverse monoids and the Wagner-Preston representation
theorem which is the inverse semigroup analogue to Cayley’s theorem.

Throughout the
paper some examples will be discussed for each of these sections, as well as a couple of
fleshed out examples, namely the Sierpinski triangle and the bicyclic monoid. The aim of
this paper will be to present these concepts so that the groundwork is there to pursue
deeper research into this topic.