Studies On Structures And Characterizations Of L~p-abundant Semigroups

In this dissertation, we study the structures and characterizations of (?)^ρ-abundant semigroups . The main idea to describe structures is by generalized Green relations and in terms of the structures of the set of idempotents in generalized regular semigroups.And we also study the translational hull and the least C-congruence of it.Rpp semigroups and wrpp semigroups are all a class of important semigroup.The structures of rpp semigroups and wrpp semigroups have been well described .Moreover, the translational hull of them is also described. In this paper .the structure and characterizationsare generalized .There are four chapters:In chapter 1, we give the introductions and preliminaries.In chapter 2, we give the structure of (?)^ρ-orthogroup. We obtain the band-like extension and the semi-spined product structure of it,The main results are given in follow:Theorem 2.2.1 Let S be a semigroup.Then the following statements are equivalent:(i)There existsρ∈(?)C(S) such that S is an (?)^ρ-orthogroup. and the following condition is satisfied: (C1) (?)x,y∈S,xρy(?)xeρye, for every e∈E(S).(ii)S≌[Y;S_α=I_α×T_α×Λ_α], where S_αis a rectangular unipotent semigroup,and there existsρ'_α∈(?)C(T_α) such that T_αis p'_α-left cancellative, E(S) is a band, and the following condition is satisfied: (C2) for every a,b∈T_α,(i,λ)∈I_α×Λ_α,if aρ'_αb,then[(i,α,λ)(j,1_Tβ,μ)]P_{Tβρ'β}[(i,b,λ)(j,1_Tβ,μ)]P_Tβfor anyβ≤α,(j,μ)∈I_β×Λ_β,(iii)S can be described as a band-like extension B[Y;T_α;I_α;Λ_α;ξ_α,βη_α,β] of some semigroup [Y;T_α], and there existsρ'_α∈(?)C(T_α) such that T_αisρ_'α-left cancellative,E(S) is a band, and the following condition is satisfied: (C3) if aρ'_αb for every a,b∈T_α, then a·1_{Tβρ'β}b·1_Tβ for anyβ≤α.Theorem 2.2.3 Let S be a semigroup.Then the following statements hold: (1) There existsρ∈(?)C(S) such that S is a rectangular (?)^ρ-orthogroup if and only if S is a rectangular unipotent semigroup I×T×Λ, and there existsρ'∈(?)C(T) such that T isρ'-left cancellative; in particular, there existsρ∈(?)C(S) such that S is a left(right) zero (?)^ρ-orthogroup if and only if S is a left(right) unipotent semigroup (I×T)(T×Λ), and there existsρ'∈(?)C(T) such that T isρ'-left cancellative.(2) There existsρ∈(?)C(S) such that S is a (?)^ρ-orthogroup in which E(S) is a semilattice and the condition (C1) is satisfied if and only if S≌[Y; T_α],where T_αis a unipotent semigroup ,and there exists p'_α∈(?)C(T_α) such that T_αisρ'_α-left cancellative, E(S) is a band, and the following condition is satisfied:a,b∈T_α, ap'_αb(?)a·1_{Tβρ'βp}b·1_Tβ,(?)β≤β,(3) There existsρ∈(?)C(S) such that S is a right(left) regular (?)^ρ-orthogroup which satisfies (C1) if and only if S is a right(left) band-like extension of some semigroup [Y; T_α], where T_αis a unipotent semigroup ,and there existsρ'_α∈(?)C(T_α) such that T_αisρ'_α-left cancellative, E(S) is a band, and (C3) is satisfied.(4) There existsρ∈(?)C(S) such that S is a regular (?)^ρ-orthogroup which satisfies (C1) if and only if there exist a left regular (?)^ρ1-orthogroup S₁ and right regular (?)^ρ2-orthogroup S₂ which satisfy (C1) such that S≌S₁×_TS₂, where S₁ and S₂ have the same branch T andρ_i∈(?)C(S_i)(i=1,2).(5) There existsρ∈(?)C(S) such that S is a left(right) quasi-normal (?)^ρ-orthogroup which satisfies (C1) if and only if there exist a left(right) regular (?)^ρ1-orthogroup S₁ and a right(left) normal band B such that S≌S₁×_YB, whereρ₁∈(?)C(S₁).(6) There existsρ∈(?)C(S) such that S is a (left,right) normal (?)^ρ-orthogroup which satisfies (C1) if and only if there exists a semigroup [Y; T_α] and a (left,right) normal band B such that S≌T×_YB,and there exists p'_α∈(?)C(T_α) such that T_αisρ'_α-left cancellative, E(S) is a band, and (C3) is satisfied.(7) There existsρ∈(?)C(S) such that S is a right(left) semiregular (?)^ρ-orthogroup which satisfies (C1) if and only if S is a right(left) band-like extension of a left(right) regular (?)^ρ1-orthogroup S₁ satisfying (C1), whereρ₁∈(?)C(S₁).(8) There existsρ∈(?)C(S) such that S is a right(left) seminormal (?)^ρ-orthogroup which satisfies (C1) if and only if S is a right (left) band-like extension of a left (right) normal (?)^ρ1-orthogroup S₁ satisfying (C1), whereρ₁∈(?)C(S₁).Theorem 2.3.2 Let T=[Y;T_α],where T_αis a unipotent semigroup.And there exists ρ'_α∈(?)C(T_α) such that T_αisρ'_α-left cancellative, E(T) is a band. For every elementα∈Y,we associateαwith two non-empty sets I_αandΛ_α,such that I_α∩I_β=Λ_α∩Λ_β=(?) wheneverα≠βin Y. Denote I=(?)I_α,Λ=(?)Λ_αand S_α= I_α×T_α×Λ_α. We define the mappings bysuch that the following statements hold:(L₁) If (i,x,λ)∈S_αand j∈I_β,then(i,x,λ)^#j∈I_αβ.(R₁) If (i,x,λ)∈S_αandμ∈Λ_β,thenμ(i,x,λ)^*∈Λ_αβ.(L₂) in (L₁), Ifα≤β,then (i,x,λ)^#j=i.(R₂) in (R₁), Ifα≤β,thenμ{i,x,λ)^*=λ.(L₃) If (i,x,λ)∈S_αand (j,y,μ)∈S_β,then (i,x,λ)^#(j,y,μ)^?= ((i.x,λ)^#j,xy,λ(j,y,(R₃)If (i,x,λ)∈S_αand(j,y,μ)∈S_β,then(i,x,λ)^*{j,y,μ)^*=((i,x,λ)^#j,xy,λ(j,y,μ)^*)^*.(P)(?)β≤α(α,β∈Y),aρ'_αb(?)(a1_Tβ)ρ'_β(b1_Tβ.Then the set S=(?)S_αforms a semigroup of which E(S) =(?)(I_α×{1_Tα}×Λ_α),with respect to the following operation defined by(i,x,λ)(j,y,μ)=((i,x.λ)^#j,xy,λ(j,y,μ)^*)And there existsρ∈(?)C(S) such that S is a (?)^ρ-orthogroup which satisfies (C1). Conversely, each (?)^ρ-orthogroup satisfying (C1) can such be constructed.Theorem 2.3.5 Let S be a semigroup.Then the following statements hold:(1) There existsρ∈(?)C(S) such that S is a rectangular (?)^ρ-orthogroup if and only if S is a rectangular unipotent semigroup I×T×Λ, and there existsρ'∈(?)C(T) such that T isρ'-left cancellative; in particular, there existsρ∈(?)C(S) such that S is a left(right) zero (?)^ρ-orthogroup if and only if S is a left(right) unipotent semigroup (I×T)(T×Λ), and there existsρ'∈(?)C(T) such that T is p'-left cancellative. (2) There existsρ∈(?)C(S) such that S is a (?)^ρ-orthogroup in which E(S) is a semilattice and the condition (C1) is satisfied if and only if S≌[Y;T_α],where T_αis a unipotent semigroup ,and there existsρ'_α∈(?)C(T_α) such that T_αisρ'α-left cancellative, E(S) is a band, and the following condition is satisfied:a.b∈T_α, aρ'_αb(?)a1_{Tβρ'β}b1_Tβ,(?)β≤α.(3) There existsρ∈(?)C(S) such that S is a right(left) regular (?)^ρ-orthogroup which satisfies (C1) if and only if S is a right (left) semi-spined product of some semigroup [Y;T_α], where T_αis a unipotent semigroup. And there existsρ'_α∈(?)C(T_α) such that T_αisρ'_α-left cancellative, E(T) is a band.(4) There existsρ∈(?)C(S) such that S is a regular (?)^ρ-orthogroup which satisfies (C1) if and only if there exist a left regular (?)^ρ1-orthogroup S₁ and right regular (?)^ρ2-orthogroup S₂ which satisfy (C1) such that S≌S₁×_TS₂,where S₁ and S₂ have the same branch T andρ_i∈(?)C(S_i)(i=1,2).(5) There existsρ∈(?)C(S) such that S is a left(right) quasi-normal (?)^ρ-orthogroup which satisfies (C1) if and only if there exist a left(right) regular (?)^ρ1-orthogroup S₁ and a right(left) normal bandΛsuch that S≌S₁×_YΛ, whereρ₁∈(?)C(S₁).(6) There existsρ∈(?)C(S) such that S is a (left,right) normal (?)^ρ-orthogroup which satisfies (C1) if and only if there exists a semigroup [Y;T_α] and a (left,right) normal band B such that S≌T×_YB,and there existsρ'_α∈(?)C(T_α) such that T_αisρ'_α-left cancellative, E(T) is a band.(7) There existsρ∈(?)C(S) such that S is a right(left) semiregular (?)^ρ-orthogroup which satisfies (C1) if and only if S is a right(left) semi-spined product of a left(right) regular (?)^ρ1orthogroup S₁ satisfying (C1),whereρ₁∈(?)C(S₁).(8) There existsρ∈(?)C(S) such that S is a right(left) seminormal (?)^ρ-orthogroup which satisfies (C1) if and only if S is a right(left) semi-spined product of a left(right) normal (?)^ρ1-orthogroup S₁ satisfying (C1),whereρ₁∈(?)C(S₁).In chapter 3, we give the definition of a type A-(?)^ρ-abundant semigroup and describe the translational hull of it.The main results are given in follow:Definition 3.2.1 Let S is a semigroup. S is a type A-(?)^ρ-abundant semigroup, if S is a strong C-(?)^ρ-abundant semigroup and the following condition(M) is satisfied: eρf(?)e=(?)f,for every e, f∈E(S).Theorem 3.2.11 Let S is a type A-(?)ρ-abundant semigroup.Then there exists (?)∈ (?)C(Ω(S)) such thatΩ(S) is a type A-(?)abundant semigroup.In chapter 4, we give the smallest C-congruence on (?)^ρ-abundant semigroup. The main result is given in follow:Theorem 4.2.3 Letσ=(?)σ_α.Then:(1)σis a congruence on the semigroup S.(2) S/σis a semilattice Y ofρ'_α-left cancellative semigroups and E(S/σ) is central.(2)σis the smallest congruence on the semigroup S which satisfies(2).