Several Research Of Two Classes Of Generalized Regular Semigroups

Ill this paper.We mainly study two classes of generalized regular semigroups,some properties and structure theorems of such semigroup are given.The main idea is to describe properties and structures of generalized regular semigroups by gener-alized Green relations on generalized regular semigroup.There are three chapters,the main contest are given in follow:In the first chapter,we study congruence and properties of strongly U-HU-abundant semigroups,we give three biggest congruences contained in LU,RU,HU respectively,many equivalent characterizations are given by the three congruences.Finally,we characterize some properties of strongly U-HU-abundant semigroups in term of idempotents.The main results are given in follow:Theorem 1.2.2 Let S be strongly U-LU-abundant semigroups,thenμ(?)U={(a,b)∈S×S|(?)u∈U1,(ua,ub)∈LU}={(a,b)∈S×S|(?)u∈U1,(ua)*=(ub)*}.is biggest congruence contained in LU.Theorem 1.2.3 Let S be strongly-V-RU-abundant semigroups,thenμ(?)U={(a,b)∈S×S|(?)u∈U1,(au,bu)∈RU}={(a,b)∈S×S|(?)u∈U1,(au)+=(bu)+}.is biggest congruence contained in RU.Theorem 1.2.4 Let S be stronglh U-HU-abundant semigroups,thenμ={(a,b)∈S×S|(?)u∈U1,(ua,ub)∈LU,(au,bu)∈RU}={(a,b)∈S×S|(?)u∈U1,(ua)*=(ub)*,(au)+=(bu)+}.is biggest congruence contained in HU.Theorem 1.2.8 Let S be strongly U-LU-abundant semigroups,then the follow statements are equivalent:(1)S=U={uμ(?)U|u∈U};(2)μ(?)U=LU;(3)For(?)u∈U,a∈S,(au,ua)∈LU.Theorem 1.2.9 Let 5 be strongly U-RU-abundant semigroups,then the follow statements are equivalent:(1)S=U={Uμ(?)U|u∈U];(2)μ(?)U=RU;(3)For(?)u∈U,a∈S,(au,ua)∈RU.Theorem 1.2.10 Let S be strongly U-HU-abundant semigroups,then(1)For(?)a,b∈S,a*=a+,b*=b+;(2)LU=RU=HU;(3)EachHU-class of S only contain an idempotont in U;(4)U is center of S.Theorem 1.2.12 Let s be strongly U-LU-abundant semigroups and if S is strongly U-LU-abundant semigroups,then for(?)a∈S,a*=a*.Theorem 1.2.13 Let S be strongly U-RU-abundant semigroups and if S is strongly U-RU-abundant semigroups,then for(?)α∈S,a+=a+.Theorem 1.2.14 Let S be strongly U-HU-abundant semigroups and if S/μis strongly U/μ-HU/μ-abundant semigroups,then for(?)a∈S,(aμ)*=a*μ,(aμ)+=a+μ.Theorem 1.2.15 Let S be strongly U-(?)-abundant semigroups.If S is strongly U-(?)-abundant semigroups,then S is right basic.Theorem 1.2.16 Let S be strongly U-(?)-abundant semigroups.If S is strongly U-(?)-abundant semigroups,then S is left basic.Theorem 1.2.17 Let S be strongly U-(?)-abundant semigroups.If S/μ.is strongly U/μ-(?)-abundant semigroups,then S/μ is basic.In the second chapter,we give the description of the structure of tvpe A-(?)-abundant semigroups.Firstly,we give the definition of type A-(?)-abundant semigroup.Secondly,we prove that the type A-(?)-abundant semigroups’ transla-tions hull is still the type A-(?)一abundant semigroups’ translations hull.The main results are given in follow:Theorem 2.2.9 The type A-(?)-abundant semigroups’ translations hull is still the type A-(?)-abundant semigroups’translations hull.Corollary 2.2.10 The type A-(?)-abundant semigroups’ translations hull is still the type A-(?)-abundant semigroups’ translations hull.In the third chapter,we define the LR-C-good B-quasi-Ehresmann semigroups and get the structure of LR-C-good B-quasi-Ehresmann semigroups.The results are given in follow:Theorem 3.2.1 Let S be a semigroup.Then S(B)is a LR-C-good B-quasi-Ehresmann semigroup for some B C E(S)if and only if S is isomorphic to a spined product Si × Tφψ,S2 of a left C-goodB-quasi-Ehresmmann semigroup S1=[Y;Iα×Tα]and a right C-goodB-quasi-Ehresmann semigroup S2=[Y;Tα×Aα]having a common C-good B-quasi-Ehresmann semigroup component T=[Y:Tα]with respect to the semigroup homomorphisms φ:(i,x)(?)x,for(i,x)∈Si and≦:(x,λ)(?)x,for(x,λ)S2,and B(S)C(C(B(S1))x B(S2))U(B(S1)×C(B(S2))).Where B(S)=∪αY((Iα×{1Tα)×({1Tα}×Λα)),B(S1)-∪α∈Y((Iαx{1Tα}):B(S2)=∪α∈Y({1Tα}×Aα)and C(B(Si))is center of B(Si),i-1,2.Theorem 3.2.2 Let T=[Y;Tα]be a C-goodB-quasi-Ehresmann semigroup,I=[Y;Iα]is a left regular band and A=[Y;Λα]is a right regular band.If the following mappingsξ:∪α∈Y(Iα×Tα)→τl(I),(i,x)→(i,x)#η:∪α∈Y(Tα×Λα)→τr(Λ),(x,λ)→(x,λ)*Satisfying the following pairs of conditions:(L1)if(i,x)∈ Iα × Tα and j∈Iβ,then(i,x)#j∈Iαβ;(R1)if(x,λ)∈Tα×Λα and μ∈Λβ,then μ(x,λ)*∈Λαβ;(L2)ifα≤βin(L1),then(i,x)#j=i;(R2)ifα≤βin(R1),then μ(x,λ)*=λ;(L3)if(i,x)∈Iα×Tαand(j,y)∈Iβ×Tβ,then(i,x)#(j,y)#=((i,x)#j,xy)#;(R3)if(x,λ)∈Tα×Λαand(y,μ)∈Tβ×Λβ,then(x,λ)*(y,μ)*=(xy,λ(y,μ)*)*;(P)if i∈Iα,λ∈Λα,then(?)β≤α,(?)j∈Iβ,(i,1Tα)#j=j(according to(L2)to|Iα|=1)or(?)β≤α,(?)μ∈Λβ,μ(1Tα,λ)*=μ(according to(R2)to |Λα|=1).Then S(B)=∪α∈Y(Iα×Tα×Λα)forms a LR-C-goodB-quasi-Ehresmann scrnigroup,where B=∪α∈Y(Iα×{1Tα} ×Λα),under the following multiplication:(i,x,λ)(j,y,μ)=((i,x)#j,xy,λ(y,μ)*).Conversely,every LR-C-goodB-quasi-Ehresmann semigroup can be constructed in the above manner.Theorem 3.2.3 Let T=[Y;Tα]-be a C-goodB-quasi-Ehrresmann semigroup,let Iαand Λα be two non-empty sets such that Iα∩Iβ=(?)=Λα∩Λβ(α≠β)for(?)α∈Y.Do direct product Pa=Iα×Tα and Qα=Tα×Λα(α∈ Y).Denote S=∪α∈Y(Iα×Tα×Λα)and B=∪α∈Y(Iα×{1Tα}×Λα).Now for(?)α,γ∈Y,γ≤α,define the mappingsψα,γ:Pα→τl(Iγ),(i,x)→ψα,γ(i,x)φα,γ:Qα→τr(Λγ),(x,λ)→φα,γ(x,λ)satisfying the following pairs of conditions:(L1)if(i,x)∈Pα and j∈Iα,then ψα,α(i,x)j=i;(R1)if(x,λ)∈Qαand μ∈Λα,then μφα,α(x,λ)=λ;(L2)if(i,x)∈Pαand(j,y)∈Pβ:then ψα,αβ(i,x)ψβ,αβ(j,y)=〈ψα,αβ(i,x)ψβ,αβ(i,y)〉;(R2)if(x:λ)∈Qαand(y,μ)∈Qβ,φα,αβ(x,λ)φβ,αβ(y,μ)=〈φα,αβ(x,λ)φβ,αβ(y,μ)〉;(L3)if δ≤αβ in(L2),then ψαβ,δ(k,xy)=ψβ,αδ(i,x)ψβ,δ(j,y),where k=〈ψα,αβ(i,x)ψβ,αβ(i,y)〉;(R3)if δ≤αβin(R2),then φαβ,δ(xy,v)=φαδ(x,λ)φβ,δ(y,μ),where v=〈φα,αβ(x,λ)φβ,αβ(y,μ)〉;(P)if i ∈Iα,λ∈Λβ,then(?)γ≤α,ψα,γ(i,1τα)=εIγ,(εIγ is identity mapping on Iγ)(according to(Li)to |Iα|=1)or(?)γ≤α,φα,γ(1Tα,γ)=εΛγ,(εΛγ is unit mapping on Λγ)(according to(R1)to |Λα|=1).Then S(B)=∪α∈Y(Ia x Tα×Λα)forms a LR-C-goodB-quasi-Ehresmann semigroup,where B=∪α∈Y(Iα×{1T}×Λα),under the following multiplication:(i,x,λ)(j,y,μ)=(〈ψα,αβ(i,x)ψβ,αβ(j,y)〉,xy,〈φα,αβ(x,λ)φβ,αβ(y,μ)〉).Conversely,every LR-C-goodB-Ehresmann semigroup can be constructed in the above manner.Corollary 3.2.4 Let T=[Y;Tα]be a C-Ehresmann semigroup.I=[Y;Iα]is a left regular band and Λ=[y;Λα]is a right regular band.If the following mappingsξ:∪α∈Y(Iα×Tα)→τl(I),(i,x)→(i,x)#η:∪α∈Y(Tα×Λα)→τr(Λ),(x,λ)→(x.λ)*satisfying the following pairs of conditions:(L1)if(i,x)∈Iα;x Ta and j∈Iβ,then(i,x)#j∈Iαβ;(R1)if(x,λ)∈Tα×Λαand μ∈Λβ,then μ(x,λ)*∈Λαβ;(L2)if α≤β in(L1),then(i.x)#j=i;(R2)ifα≤βin(R1),then μ(x,λ)*=λ;(L3)if(i,x)∈Iα×Tα and(j,y)∈Iβ×β,then=(i,x)#(j,y)#=(i,x)#j,xy)#;(R3)if(x,λ)Tax Aα and(y,μ)∈Tβ×Λβ,then(x,λ)*(y,μ)*=(xy,λ(y,μ)*)*;(P)if i ∈Iαλ∈Λα,then(?)β≤α,(?)j ∈Iβ,(i,1Tα)#j=j(according to(L2)to|Iα|=1)or(?)β≤α,(?)μ∈Λβ,μ(1Tα,λ)*=(according to(R2)to |Λα|=1).Then S(B)=∪α∈Y(Iα×Tα×Λα)forms a LR-C-Ehresmann semigroup,where B=∪α∈Y(Iα×{1Tα}×Λα),under the following multiplication:(i,x,λ)(j,y,μ)=((i,x)#j,xy,λ(y,μ)*).Gonversely,every LR-C-Ehresmann semigroup can be constructed in the above manner.Corollary 3.2.5 Let T=[Y:Tα]be a C—Ehresmann semigroup,let Iαand Λα be two non-empty sets such that Iα∩ Iβ=(?)=Λα∩ Λβ(α≠β)for(?)α∈Y.Do direct product Pα=Iα × Tαand Qα=Tα×Λα(α∈Y).Denote S=∪α∈Y(Iα×Tα×Λα)and B=∪α∈Y(Iα×{1Tα}×Λα).Now for(?)α,γ∈Y,γ≤α,define the mappingsψα,γ:Pα→τl(Iγ),(i,x)→ψα,γ(i,x)φα,γ:Qα→τr(Λγ),(x,λ)→φα,γ(x,λ)satisfying the following pairs of conditions:(L1)if(i.x)∈Pα and j ∈α,then ψα,α(i,x)j=i;(R1)if(x,λ)∈Qαand μ∈Λα,then μφα,α(x,λ)=λ;(L2)if(i,x)∈ Pa and(j,y)∈Pβ,then ψα,αβ(i,x)ψβ,αβ(j,x)=〈ψα,αβ(i,x)ψβ,αβ(j,x)〉);(R2)if(x,λ)∈ Qα and(y.μ)∈ then φα,αβ(x,λ)φβ,αβ(y,μ)=〈φα,αβ(x,λ)φβ,αβ(y,μ)〉);(L3)if δ≤αβ in(L2),then ψαβ,δ(k,xy)=ψβ,δ(j,y),where k=〈ψα,αβ(i,x)ψβ,αβ(j,y)〉;(R3)if δ≤αβ in(R2),then φαβ,δ(xy,v)=φα,δ(x,λ)φβ,δ(y,μ),where v=〈φα,αβ(x,λ)φβ,αβ(y,μ)〉;(P)if i ∈Iα,λ∈Λα,then(?)γ≤α,ψα,γ(i,1Tα)=εIγ,(εIγ is identity mapping on Iγ)(according to(L1)to |Iα|=1)or(?)γ≤α,φα,γ(1Tα,λ)=εΛγ,(εΛγ is unit mapping on Λγ)(according to(R1)to |Λα|=1).Then S(B)=∪α∈Y(Iα×Tα×Λα)forms a LR-C-Ehresmann semigroup,where B=∪α∈Y(Iα×{}1Tα} × Λα),under the following multiplication:{i,x,λ)(j,y,μ)-(〈ψα，αβ(i,x)ψβ,αβ(j,y)〉,xy,〈φα,αβ(x,λ)φβ,αβ(y,μ)〉).Conversely,every LR-C-Ehresmann semigroup can be constructed in the above manner.