Inverse semigroups and inverse categories

The theory of inverse semigroups forms a major part of semigroup theory. This theory has deep connections with important mathematical disciplines, not only classical ones such as geometry, functional analysis, number theory, but also with more recent theories: the theory of algorithms, graph theory, the mathematical theory of automata, etc. The importance of inverse semigroups, first and foremost, is that they form an abstract class of algebraic structures which are isomorphic to semigroups of partial bijections. This thesis is organized in two parts, inverse semigroups in Part 1, and inverse categories (that arises if we apply a basic property of inverse semigroups to morphisms of a category) in Part 2. In the first chapter of my thesis, I set out the basic properties of semigroups and inverse semigroups: this includes the isomorphism theorem for semigroups, the algebraic properties of inverse semigroups connected with inverses, idempotent elements, Green\u27s relations, etc. The examples presented at the end of the first chapter include the inverse semigroups of partial bijections, the free monogenic inverse semigroup and an inverse semigroup-like set which is an analogue of group-like sets. The last example was used in our paper [9], and was one of the examples suggested the need for an inverse semigroup-like set theory. The theory of inverse categories, a natural generalization of the theory of inverse monoids, may be regarded as the theory of partial isomorphisms. In chapter two of this thesis, I present the basic properties of inverse categories which are analogous to the properties of inverse semigroups. Four examples of inverse categories are discussed (given) at the end of my thesis: the inverse category of partial bijections, the inverse category of invertible matrices, the inverse category that represent an equivalence relation, and an inverse category as a subcategory of the category of based sets and based functions