Structures And Characters On π-regular Semigroups

In this dissertation, we study the structures of (?)^*-covered r-canceletive GV-semigroups,and we extend some results of completely regular semigroups toÏ€-regular semigroups,Lastly,we give the definition of quasi-GV-semigroup. and describe the structures of the semigroups and some special subclasses.We obtain the band-like extension of these semigroups.There are four chapters:In the first chapter , we give the introductions and preliminaries.In the second chapter , we describe the structures of (?)^*-covered r-canceletive GV-semigroups.The main results are given in follow:Theorem 2.2.1 Let Y be a semilattice.and {S_α|α∈Y}be a class of completely archimedean semigroups which don't intersect each other.S=(?){S_α|α∈Y},(?)α,β∈Y,α>β,we define a morphism:The following description will be satisfied:(1)(RegS_α)θ_α,β(?)P^*(I_β)×P(M_β).(2)Ifαis not the maximal,then S_αis a rectangular band.[We now define T_α= I_α×M_α](3)If (?)α,β∈Y,α≠αβ≠β,then |S_αβ|=1.[We now define S_αβ={(?)_αβ}](4)Ifαis not the maximal orαis the maximal ,but is not the covering ofβin Y. we haveA binary operationοis defined:(?)x∈S_α,y∈S_βThen (S,ο) is a (?)^*-covered r-canceletive GV-semigroup.Conversely,each (?)^*-covered r-canceletive GV-semigroup which satisfies ifαis not the maximal,then S_αis a rectangular band can be constructed by the above statements. In the third chapter,we extend some results of completely regular semigroups toÏ€-regular semigroups.The main results are given in follow:Theorem 3.2.1 The following conditions on GV-semigroups are equivalent:(1)S is a left regular quasi-orthogroup:(2)S is a semilattice of the nil-extension of left groups,and E(S)is a band;(3)(?)^* is a semilattice congruence on S,and E(S)is a band;(4)S satisfies the identity r(a)r(x)=r(a)r(x)r(a)⁰.Theorem 3.2.3 Let S be quasi-band,and S is r-canceletive,then the following conditions are equivalent:(1)S is a left regular quasi-band:(2)S is a semilattice of the nil-extension of left zero semigroups;(3)R^*=ε.Theorem 3.2.5 Let S be aÏ€-regular semigroup,then the following conditions are equivalent:(1)S is the nil-extension of rectangular bands:(2)(?)a,b∈S ,r(a)=r(a)br(a):(3)(?)a,b,x∈S ,r(a)r(b)=r(a)xr(b).In the fourth chapter,we give the definition of quasi-GV-semigroup, and describe the structures of the semigroups and some special subclasses.We obtain the band-like extension of these semigroups.The main results are given in follow:Theorem 4.2.1 S is the semilattice ofÏ€-group plank with identity,and {I_α×{1_Tα}×Λ_α|α∈Y} is a band if and only if S is the band-like extension of T= [Y;¹T_α],and {1_Tα|α∈Y} is a band.Theorem 4.2.2 Let S be a semigroup.Then the following statements hold:(1)S is a rectangular quasi-GV-semigroup if and only if S is aÏ€-group plank with identity:in particular,S is left (right) zero quasi-GV-semigroup if and only if S is the direct product of a left(right) zero band and aÏ€-group with identity.(2)S = [Y; I_α×¹T_α×Λ_α] is a quasi-GV-semigroup in which {I_α×{1_Tα}×Λ_α|α∈Y} is a semilattice if and only if S is a semilattice ofÏ€-group with identity.(3)S= [Y; I_α×¹T_α×Λ_α] is a right regular quasi-GV-semigroup if and only if S is a right band-like extension of [Y;¹T_α],¹T_αis aÏ€-group with identity and {1_Tα|α∈Y} is a band. (4)S=[Y;I_α×¹T_α×Λ_α] is a left regular quasi-GV-semigroup if and only if S is a left band-like extension of [Y;¹T_α],and {1_Tα|α∈Y} is a band.(5)S=[Y;I_α×¹T_α×Λ_α] is a regular quasi-GV-semigroup if and only if there are a left regular quasi-GV-semigroup S₁ and a right regular quasi-GV-semigroup S₂,which have the same [Y;¹T_α] branch T.such that S≌S₁×(?)S₂,and {1₍T_α|α∈Y} is a band.(6)S=[Y;I_α×¹T_α×Λ_α] is a left (right) quasi-normal quasi-GV-semigroup if and only if there are a left (right) regular quasi-GV-semigroup S₁= [Y;I_α×T_α] and a right (left) normal band B =[Y;B_α],such that S≌S₁×(?)B,and {1_Tα|α∈Y} is a band.(7)S=[Y;I_α×¹T_α×Λ_α] is a (left, right) normal quasi-GV-semigroup if and only if there are T = [Y;¹T_α] which is the semilattice ofÏ€-group plank with identity and a (left,right)normal band B = [Y; B_α],such that S(?)T×_Y B,and {1_Tα|α∈Y} is a band.(8)S=[Y;I_α×¹T_α×Λ_α] is a right semiregular (right seminormal) quasi-GV-semigroup if and only if S is a right band-like extension of a left regular(left normal) quasi-GV-semigroup,and {1_Tα|α∈Y} is a band.(9)S=[Y;I_α×¹T_α×Λ_α] is a left semiregular (left seminormal) quasi-GV-semigroup if and only if S is a left band-like extension of a right regular(right normal) quasi-GV-semigroup S₁ = [Y;¹T_α×Λ_α;(?)_α,β],and {1_Tα|α∈Y} is a band.