Studies On Some Generalized Completelty Regular Semigroups

In this dissertation, we first define the superabundant cryptogroup and give its characterization; and then we characterize the eventually orthodox super rpp semigroup. The main results are given in follow.In Chapter 1, we give the introduction and preliminaries.In Chapter 2, we give a definition of superabundant cryptogroup and discuss the structure of superabundant cryptogroup. The main results are given in follow. Define 2.1 The semigroup S is called a superabundant cryptogroup, if the following conditions are satisfied:ⅰ)S = [Y: S_α], where S_α=M(B_α,T_α,P_α) is a Rees matrix semigroup on cancellative monoid T_α. and P_αis normalized in (?)∈B_α:ⅱ)For any (a, h)∈S_α. (b. g)∈S_β.ⅲ) H^* is a congruence on the semigroup S.Theorem 2.2 LetB = (Y:B_α) be a band. where Y is a semilattice. For everyα∈Y, letS_α= M(B_α. T_α; P_α) be a Rees matrix semigroup on the cancellative monoid T_α. l_αthe identity element of T_α, and the sandwich matrix P_αnormalized in the fixed (?)∈B_α, letwhereα,β∈Y andα≥β,α∈B_α. For anyα,β∈Y,α≥β, defined a homomorphismθ_α,β:T_α→T_βsuch that for anyα,β,γ∈Y.α≥β≥γ, the following condition are satisfied: for any x = (a, g)∈S_α, y = (b, h)∈S_β, on S=∪_α∈Y S_αthe operations are defined as follows:Then S is superabundant cryptogroup,conversely every superabundant cryptogroup can be so constructed.In chapter 3, we define the eventually orthodox super rpp semigroup, then we study the characterization of the eventually orthodox super rpp semigroup. The main results are given in follow.Theorem 3.9 The following statements are equivalent for a semigroup S:ⅰ)S is an eventually orthodox super rpp semigroup:ⅰ)S is an expasion S = [S; T:ξ] of an orthodox super rpp semigroup T = [Y: S_α];ⅲ)S is a semilattice Y of expasions T_α= [T_α, S_α:ξ_α] of left cancellative plank S_α, and for any a∈T_α, b∈T_β, ab =αξ_α,bξ_β∈Sα_β, and for anyβ≤α∈Y. (i, x.λ)∈S_αand (j. 1_Tβ,μ). (k. 1_Tβ.V∈S_? that satisfies the condition:Corollary 3.10 The following statements are equivalent for a semigroup S:ⅰ)S is eventually C-rpp semigroup:ⅱ)S is an expasion S = [S; T:ξ] of C-rpp semigroupT = [Y; T_α]:ⅲ)S is a semilattice Y of expasions S_α= [S_α, T_α:ξ_α] of left cancellative monoid T_α: and for any a∈S_α, b∈S_β, ab =αξ_αbξ_?∈S_αβ: and for anyβ≤α∈Y, (i, x,λ)∈S_αand (j, 1_Tβ,μ), (k, 1_Tβ, v)∈S_βthat satisfies the condition:Corollary 3.11 The following statements are equivalent for a semigroup S:ⅰ)S is eventually left C-rpp semigroup;ⅱ)S is an expasion S = [S; T;ξ] of left C-rpp semigroup T = [Y; I_α×T_α]; ⅲ)S is a semilattice Y of expasions S_α= [S_α, I_α×T_α;ξ_α] of the direct product I_α×T_α, where I_αis a left zero band, the T_αis the left cancellative monoid semigroup,and (?)a∈S_α, b∈S_β, ab=aξ_αbξ_β∈S_αβ, and for anyβ≤α∈Y, (i, x)∈S_αand (j, 1_Tβ), (k, 1_Tβ)∈S_βthat satisfies the condition:Corollary 3.12 The following statements are equivalent for a semigroup S:ⅰ)S is eventually right C-rpp semigroup;ⅱ)S is an expasion S = [S;T;ξ] of right C-rpp semigroup T = [Y;T_α×(?)_α];ⅲ)S is a semilattice Y of expasions S_α= [S_α, T_α×(?)_α;ξ_α] of the direct product T_α×(?)_α, where T_αis the left cancellative monoid semigroup, (?)_αis a right zero band, and (?)a∈S_α; b∈S_βab = aξ_αbξ_β∈S_α(?).