Studies About Some Generalized Regular Semigroups

In this paper, we mainly give the definitions of some generalized regular semi-groups,and some properties and some structure theorems of such semigroups aregiven.The main idea is to describe structures of generalized regular semigroups bygeneralized Green relations in generalized regular semigroups.There are five chap-ters,the main contest are given in follow:In the first chapter, we give the introductions and preliminaries.In the second chapter§we give the description of the structure of broad semi-groups with normal medial idempotents.Secondly,we obtain some characterizationsof such semigroups.Thirdly,we establish the structure of broad semigroups with nor-mal medial idempotents. The main results are given in follow:Theorem2.2.1Let S be a broad semigroup,u is a normal media idempotentof S,so(1)(x∈S, e∈(R|-)_x∩E(S), f∈(L|-)_x∩E(S))x=eux=xuf;(2)uS(Su, uSu)is quasi appropriate semigroup§andE(uS)=u(E|-)=uE(E(Su)=(E|-)=Eu, E(uSu)=u(E|-)=uEu).Theorem2.3.5Let S be a broad semigroup with normal medial idempotents,(E|-)=<E>is the Regular subsemigroup which consisted by idempotents,soS=～W ((E|-), uSu).In the third chapter, we give some properties about quasi abundant Semigroupand conform the existence of the minimum adeuate good congruence.The main re-sult is given in follow:Theorem3.2.1Letφ: S→T is a morphism of semigroups.Then the follow-ing statements are equivalent:(1)φis a good morphism;(2)(?)a∈S,,and u∈(L|-)_a~U, v∈(R|-)_a~U,§which satisfies the condion aφLUφuφ, aφRUφvφ.Theorem3.2.5γ is the minimum adequate good congruence on quasi-adquate abundant semigroup S. Theorem3.2.9η is a congruence if and ony if for all a,b∈S: aU(a*)U(b+)b∈u((ab+)abU((ab)*).In the fourth chapter:we give the description of the structure of orthodox LU abundant semigroups.Firstly,we give the definition of orthodox super LU-abundant semigroups.Secondly,we obtain some characterizations and the band-like extension of orthodox super LU-abundant semigroups.The main result are given in follow:Theorem4.1.6Let S be a semi-lattic Y of a rectangular monoid Sα Iα×Tα×Λα∈(α∈Y),S=[Y:Iα×Λα(α∈Y)]and U=I×{1T}×Λ is a rectangular band.,for any (i,x,λ),(i’,x,λ’)∈Sα and V(j,y,μ),(j’,y,μ)∈Sβ,we have [(i,x,λ)(j,y,μ)]PTα,β=[(i’,x,λ’)(j’,y,μ’)]PTα.βTheorem4.2.1Let S be a semigroup.Then the following statements hold:(1)S is a strongly-LU-abundant semigroups and U=I×{1T}×Λ is a rect-angular band；(2)S is a orthodox super LU-abundant semigroups and U=I×{1T}×Λ is a rectangular band；(3)S is equivalent to a rectangular unipotent semigroup,I×T×Λ.Theorem4.2.2Let S be a semigroup.Then the following statements hold:(1)S is a orthodox super LU-abundant semigoup；(2)S=[Y:Sα=Iα×Tα×Λα](α∈Y),Sα is a rectangular unipotent semigroup,U=Uα∈YUα is a band.And∨(i,a,λ)∈Sα,(j,μ)∈Iβ×Λβ,(κ,v)∈Iγ×Λγ.[(i,a,λ)(j,1Tβ,μ)]PIαβ=[(i,a,λ)(k,1Tγ,v)]PIα (?)[(i,1Tα,λ)(j,1Tβ,μ)]PIαβ.=[(i,1Tα,λ)(κ,1Tγ,v)]PIαγ.In the fifth chapter,we give the description of the structure of type A LU-abundant semigroups.Firstly,we give the definition of type A-LU-abundant semigroups.Secondly.we give a structure theorem of the type A-LU-abundant semigroups’ translations hull.The main result is given in follow:Definition5.2.4Let S be a type A-LU-abundant semigroups,if S be a LU-abundant semigroups,U is a semi-lattic,and satisfies the condion(M):(Va∈S,u∈U)ua=a(ua)u. Theorem5.2.4Let (λ, ρ)∈(S), so that (λ~u, ρ~u)∈U((S)).Theorem5.2.11If S is a type A-(L|-)~U, abundant semigroup,then the trans-lational hull of S is also a type A-(L|-)~U-abundant semigroup.