| In this dissertation, we first define the perfectly superabundant semigroup and give its characterization;and then we characterize the eventually U-liberal orthogroups. The main results are given in follow.In Chapter 1, we give the introduction and preliminaries.In Chapter 2, we give a definition of perfectly superabundant semigroup and discuss the structure of perfectly superabundant semigroup.The main results are given in follow. Define 2.1 The semigroup S is called a perfectly superabundant semigroup, if the following conditions are satisfied:i)S = [Y;S_α], where S_α = M[I_α,T_α, (?)_α;P_α] is a Rees matrix semigroup on cancellative monoid T_α, and P_α is normalized in (1_α,1'_α) ∈ I_α × (?)_α.ii)L~* 和 R~* are the congruences on the semigroup 5.iii)For all (i,a,λ) ∈ S_α, (j, b,μ) ∈ S_β,Theorem 2.3 Let I = (Y;I_α) and A = (Y;(?)_α) be a left regular band and a right regular band respectively, where Y is a semilattice. For every a ∈ Y, let S_α = M(I_α,T_α,(?)_α;P_α) be a Rees matrix semigroup on the cancellative monoid T_α, (?)α the identity element of T_α, and the sandwich matrix P_α normalized in the fixed (1_α, 1'_α) ∈ I_α × (?)_α. Letwhere α,β ∈ Y, a ≥β, and i ∈I_a,λ∈(?)_a. For any α,β ∈ Y,α≥β, defined a homomorphism , such that for any , the following condition are satisfied :for anyfor any , whereIf x = (i,g, A) G Sa,j? = (j,h,fi) € 5^, and the operation on 5 = \J Sa is denned as follows:Then S is perfectly superabundant semigroup.Conversely ,every perfectly superabundant semigroup can be so constructed.In chapter 3, we study eventually U-liberal orthogroups. The main results are given in follow. Theorem 3.10 The following statements are equivalent for a semigroup S :i)S(U) is an eventually U-liberal orthogroup for some U C E(S);ii)S(U) is an expasion 5 = [S;T;￡\ of U-liberal orthogroup T - [Y;Sa(Ua)] for someU C E{S);iii)S is a semilattice of expasions Ta — [Ta,Sa;￡a] of rectangular monoid Sa(Ua) and forany o € Ta, b e T0, ab = a^ab^0 ￡ Sa0, U= \J UQ is a band;aeY Corollary 3.11 The following statements are equivalent for a semigroup S:i)S(U) is eventually C-Ehresmann semigroup for some U C E(S);\i)S{U) is an expasion 5 = [S;T;f] of C-Ehresmann semigroupT = [Y;Ta] for some U CB(S);iii)S is a semilattice of expasions Sa = {Sa,Ta;￡a] of monoid Ta , and for any a 6 Sa,b ￡ Sp,ab = a￡ab￡i3 G Sap, (J {1^ } is a semilattice .aeY... |
