The Discussion Of Generalized Rees Matrix Semigroups

In this dissertation, we first generalize Rees matrix semigroups and study the abstract characterizations of generalized Rees matrix semigroups .Further ,The main idea is to discuss generalized Rees matrix semigroups by generalized Green relations.Rectangular groups and left groups are all very important semigroups. In Mario[2],the two semigroups all have been described.In this paper,the generalized rectangular groups and generalized left groups will be described.There are three chapters.In Chapter 1, we deal with the characterizations of generalized Rees matrix semigroups .Firstly we introduce the concept of Rees matrix semigroups without zero i.e. a semigroup M = M[T] I, Λ;P] which is an analogue of Rees matrix semigroups .Where T is a monoid,Λ and I are non-empty sets and P is a Λ× I -matris over T with entries pλi, where (λ, i) ∈ Λ × I .The multiplication on M is defined by(i,x,λ)(j,y,μ)=(i,xp_λjy,μ).moreover the abstract characterizations of three classes of Rees matrix semigroups without zero that is * * -completely simple semigroups ,^<sup>-completely simple semigroups and Δ-completely simple semigroups are provided.They are the natural generalization of completely simple semigroups .The first is defined as a semigroup S which satisfies the following conditionsi) S contains a single regular D—class;ii) S satisfies rpp and lpp conditions;iii) All elements of E(S) are primitive.The second is defined as a semigroup S which satisfies the following conditionsi) S is weakly cancellative;ii) S contains a single regular D—class;iii) S satisfies semi—rpp and semi—lpp conditions.The last is defined as a semigroup 5 which satisfies the following conditionsi) 5 is right cancellative;ii) 5 contains a single regular D—class;iii) ￡=S x 5;iv) 5 satisfies semi—Ipp condition.In Chapter 2, we study the questions of * —completely simple semigroups when E(S) is a subsemigroup of it,we give the definitions of *-rectangular group and E — * -completely simple semigroups,and prove their equivalence,furthermore ,we also give other equivalences and the homomorphism theorems of *—rectangular group .In Chapter 3, the questions of *—rectangular group when it satisfies | A |= 1 will be proved.First we give the definitions of *—left group and *—group,then we give their some corresponding equivalences.