Keywords (Extracted from title, table of contents and abstract of thesis)
monoids, semirings, ssystem, shomomrphisms, rsemidodules, rhomomorphisms, von neumann regular semiodules, sheafs 
Abstract A ring R is called regular if for each a â‚¬ R, there exists an element x â‚¬ R such that axa = a. Regular rings were introduced by von Neumann 1936, in order to clarify certain aspects of operator algebras. Since then regular rings have been very extensively studied both for their own sake, as well as for the sake of their links with operator algebras. In this thesis, we will be concerned with this important notion and some of its generalizations, from a purely algebraic point of view, in the contexts of semigroups and semirings. We will determine new characterizations of regular, weakly regular and some of the other related classes of semigroups and semirings, using algebraic and homological techniques. We will also initiate the study of sheafs for certain classes of semigroups and semirings. Throughout this thesis, which contains five chapters, S will denote a semigroup and Ssystems are representations of 5. Moreover, R will denote a semiring and Rsemimodules are nonsubtractive generalizations of modules over rings. Chapter 1 is of an introductory nature which provides basic definitions and reviews some of the background material which is needed for reading the subsequent chapters. In chapter 2, we introduce Pinjective and divisible Ssystems. We use these notions to construct an Sdivisible Ssystem, Q(A), for an Ssystem A under some conditions. We also define and characterize von Neumann regular Ssystems, and deduce several new characterizations of (von Neumann) regular monoids. In this chapter, we also study weakly regular monoids, and as a generalization of these monoids, we introduce the notion of normal Ssystems. We show that an arbitrary monoid S is weakly regular if and only if each Ssystem is normal. In chapter 3, we introduce the notion of a regular semimodule, which is analogous to the notion of (von Neumann) regular Ssystems studied in chapter 2. We characterize regular semimodules in terms of certain restricted injectivity properties, and use this characterization to obtain new characterizations of regular semirings. We also examine semiring analogs of the notions of hereditary, semihereditary and PPrings. As an application of our results in this chapter, we obtain a homological characterization of PPsemirings. We also establish a characterization theorem for projective semimodules, which is analogous to the Classical Projective Basis Theorem for projective (ring) modules. In chapter 4, we define and characterize weakly regular semirings and study some properties of their prime ideal space. In chapter 5, we construct sheafs for classes of monoids and semirings, which include regular and weakly regular monoids and semirings.
