Studies About Some Problems On Generalized Regular Semigroups

In this dissertation, we study the structures of broad semigroups and (?)-broad semigroups.The main idea is to describe structures of the generalized regular semigroups by generalized Green relations in generalized regular semigroups.Broad semigroups are a class of important semigroups.The structures of some broad semigroups have been described.In this paper,the structure will be generalized to the according to some generalized regular semigroups.There are four chapters;In the first chapter, we give the introductions and preliminaries.In the second chapter, we give the description of the structure of broad semigroups with a weak normal idempotent.Firstly,we give the definition of weak normal idempotents .Secondly, we obtain some characterizations of such semigroups. Thirdly,we establish the structure of broad semigroups with weak normal idempotents. The main results are given in follow:Theorem 2.2.3 Let S be a broad semigroup.u is a weak medial idempotent of S. Then the following statements are equivalent:(1) u is a weak normal idempotent of S;(2) (?)e∈E(S), eSe is a adequate semigroup;(3) (?)e∈E(S), E(eS) is a right normal band;(4) (?)e∈E(S), E(Se) is a left normal band;(5) E(uS) is a right normal band;(6) E(Su) is a left normal band.Theorem 2.3.4 Let (T; L, R; P) be a SQ system. If P satisfies (C5),then SQ(T; L, R; P) is a broad semigroup with a weak normal idempotent-(εl,ε,εr);otherwise, broad semigroups with weak normal idempotents can also be made as this.In the third chapter, we give the description of the structure of orthodox super (?)-broad semigroups.Firstly,we give the definition of orthodox super (?)-broad semi-groups.Secondly,we obtain some characterizations and the band-like extension of ortho-dox super (?)-broad semigroups.The main results are given in follow:Theorem 3.2.1 Let S be a semigroup.Then the following statements hold:(ⅰ)S is a strongly-(?)-broad semigroup and E(S)is a rectangular band；(ⅱ)S is a orthodox super (?)-broad semigroup and E(S)is a rectangular band；(ⅲ)s is equivalent to a rectangular unipotent semigroupΙ×Τ×ΛTheorem 3.2.2 Let S be a semigroup.Then the following statements hold:(ⅰ)S is a orthodox super (?)-broad semigroup；(ⅱ)S=[Y;Sα=Ια×Τα×Λα](α∈Y),Sαis a rectangular unipotent semigroup, E(S) is a band.And (?)(i,a,λ)∈Sα,(j,μ)∈Ιβ×Λβ,(k,ν)∈Ιγ×Λγ,[(i,a,λ)(j,1Τβ,μ)]PΙαβ=[(i,a,λ)(k,1Τγ,ν)]PΙαγ(?)[(i,1Τα,λ)(j,1Τβ,μ)]PIαβ=[(i,1Τα,λ)(k,1Τγ,ν)]PIαγTheorem 3.3.3 Let S be a semigroup.Then the following statements hold:(ⅰ)S is a orthodox super (?)-broad semigroup；(ⅱ)S can be expressed as a semilattice Y of unipotent planks-Sα=Ια×Τα×Λα(α∈Y) and E(S)is a band,which satisfies the condition:(C2)(?)β≤α(α,β∈Y),(i,χ,λ)∈Sα,(j,1Τβ,μ),(k,1Τβ,ν)∈Sβ,((i,χ,λ)(j,1Τβ,μ))ΡΙβ=((i,χ,λ)(k,1Τβ,ν))ΡΙβ(?)((i,1Τα,λ)(j,1Τβ,μ))ΡΙβ=((i,1Τα,λ)(k,1Τβ,ν))ΡΙβ.(ⅲ)S can be described as a band-like extension-B[Y;Τα,Ια,Λα;ξα,β,ηα,β] of a semigroup T=[Y;Τα] which satisfies the condition: (C3)(?)β≤α(∈Y),(i,χ,λ)∈Ια×Τα×Λα,there existsζα,β(i,χ,λ)∈Τi(Ιβ),such that:ξα,β(i,1Τα,λ)=ζα,β(i,χ,λ)*ξα,β(i,χ,λ)In the fourth chapter,we give a definition of (?)-broad semigroups,strongly-(?)-broad senigroups,type A-(?)-broad semigroups and a structure theorem of the type A-(?)-broad semigroups' translational hull.The main results are given in follow::Theorem 4.2.11 If S is a type A-(?)-broad semigroup,then the translational hall of S is also a type A-(?)-broad semigroup.