Studies On Some Generalized Regular Semigroups

In this dissertation, we give the definitions of some generalized regular semigroups and its characterization. There are three chapters. The main results are given in follow.In the first chapter . we give a definition of eventually orthodox super-wrpp semigroup and discuss the structure of eventually orthodox super-wrpp semigroups.The main results are given in follow.Definition 1.9 A semigroup S is called an eventually strongly wrpp semigroup , if for all a∈S,|L_a^(**)∩EI_a|=1.Then we denote the only element bya⁽⁺⁺⁾.An eventually strongly wrpp semigroup satisfies Ehresmann type condition,that is to say, ET-condition:((?)a,b∈S)(ab)⁽⁺⁺⁾(?)(E(S))a(⁽⁺⁺⁾b⁽⁺⁺⁾.Definition 1.10 An eventually strongly wrpp semigroup is called an eventually orthodox super-wrpp semigroup ,if E(S) is a band.and satisfies Ehresmann type condition.Theorem 1.17 S is a semigroup.The statements below are equivalent:(i)S is an eventually orthodox super-wrpp semigroup:(ii)S=[S.T:ξ] is an inflation of an orthodox super-wrpp semigroup T;(iii)S can be expressed as a semilattice Y of inflations of R- left cancellative planks S_αwhich satisfies the conditions: E(?) is a band,and for all (i,a,λ)∈S_α. (j,μ)∈I_β×Λ_β,(k,v)∈I_?×Λ_γ,and (?)a∈T_α,(?)b∈T_β,ab=aξ_αbξ_β∈S_αβ;(vi)s is an inflation of a band-like extension (?)[Y:I_α,Λ_α;ξ_α,β·η_α,β] of a C-wrpp semigroup [Y;T_α] which satisfies the condition: (?)β≤α(∈Y), (i,x,λ)∈I_α×T_α×Λ_α,there existesζ_α,β^i.x.λ∈T_l(I_β), such thatIn the second chapter , we define the inflation of cancellative plank by using partial semigroups and give a discription of isomorphism between inflations of cancellative planks.Then we discuss theξ-congruence of rectangular semiguoups.The main results are given in follow.Theorem 2.1.3 A semigroup S = [S.I×T×Λ:ξ] is a inflation of cancellative plankI×T×Λif and only if there are a isomorphism between S and a M(T: I,Λ: Q,X,(?),(?)).Theorem 2.2.1 There exists a isomorphism between∑= M(T:I,Λ:Q,X,(?),(?)) and∑' = M(T':I',Λ';Q',X',(?)',(?)') if and only if there exists a isomorphismω: T→T'? bijective maps h: I→I',k:Λ→Λ' andΩ: Q- Q.such that(?)p∈Q.(1)(?)(p)h=(?)'(pΩ);(2)(?)(p)k=(?)'(pΩ);(3)X(p)ω=)X'(pΩ)?Definition 2.3.2 A congruenceρof a semigroup S is calledξcongruence or said to satisfyξ- relation,if aρb (?)aξρbξ.Theorem 2.3.3 K = L×G×R is arectangular semiguoup.S = [S, K:ξ] is the inflation of K.r∈ε(L).N∈N(G).π∈ε(R). (?)a,b∈S.denote aξ= (i.x.λ).bξ=(J,Y,μ).Define relationρon S:thenρis aξ- congruence of S: Conversely.everyξ- congruence of S can be so constructed.In the third chapter .we give a definition of scmidirect product of C - (?) - abundant semigroup and discuss necessary and sufficient condition for the semidirect product of two monoids to be a C - (?)-abundantsemigroup.The main results are given in follow.Theorem 3.2.2 A semidirect product S×αT is a C- (?)-abundent semigroup if and only if(1)S and T are C - (?)-abundent semigroups:(2)(?)f∈E(T),s∈S(?)f^s=f;(?)e∈E(S),t∈T(?)t^e=t.Theorem 3.3.4 A wreath product SW_XT is a C -(?)^#-abundent semigroup if and only if(1)S and T are C - (?)-abundent semigroups;(2)(?)f∈E(T^X),(?)s∈S,(?)x∈X(?)f(sx) = f(x):(?)e∈E(S),t∈T^X(?) t(ex) = t(x).