On Generalized Rees Matrix Semigroups And Structures Of Some Semigroups

In this dissertation, we first generalize Rees matrix semigroups and study the abstract characterizations and the homomorphism theorems of generalized Rees matrix semigroups.Further,we discuss the characterizations of semilattice and strong semilattice of generalized Rees matrix semigroups. The main idea is to discuss generalized Rees matrix semigroups and semilattice of this kind of semigroups by generalized Green relations.Regular semigroups,paticularly completely regular semigroups are a class of very important semigroups.The structures of some orthodox completely regular semigroups have been described. In this paper,the structures of a new class of orthodox completely regular semigroups will be provided.There are three chapters.In the first chapter,we deal with the characterizations and the homomorphism theorems of generalized Rees matrix semigroups. Firstly we introduce the concept of Rees matrix semigroups without zero i.e. a semigroup M = M[T; I, (?); P) which is an analogue of Rees matrix semigroups over groups.Where T is a monoid, (?) and I are non-empty sets and P is a (?) ×I-matrix over T with entries pλi, where (λ,i) ∈ (?) × I.The multiplication on M is defined byMoreover the abstract characterizations of two classes of Rees matrix semigroups without zero that is *-completely simple semigroups and ~-completely simple semigroups are provided.They are the natural generalization of completely simple semigroups.The former is defined as a semigroup S which satisfies the following conditions(i) For all a ∈ S, Eaa≠ ^(?),(ii) For all a,b ∈ S, baL*a, abR*a. The latter is defined as a semigroup S which satisfies the following conditions(i) For all a∈S,Ea≠(?),(ii) For all a,b∈ S, baLa, abRa,(iii) S is weakly cancellative.In the end, we give the homomorphism theorems of these two classes of Rees matrix semigroups without zero.In the second chapter, we study the semilattice and the strong semilattice of *-completely simple semigroups. We mainly give the characterizations of semilattice and stong semilattice of *-completely simple semigroups by generalized Green relations L*, R*, H*,D* and L,R,H,D.In the third chapter, we describe the structures of LR-semiregular semigroups. Firstly, we give the definition of LR-semiregular semigroups i.e. an orthogroup S which E(S) is a LR-semiregular band. Secondly, we obtain the semi-spined product structure and the A-product structure of this kind of semigroups.