Studies On Some Generalized Regular Semigroups

This passage gives the definitions of some Generalized Regular Semigroups and the theorems of their structures.In Chapter 1, we give the introductions and preliminaries.In Chapter 2, we give a definition of Rees matrix semigroup on a cancellative moniod and discuss the congruence of Rees matrix semigroup on a cancellative moniod. The main results are given in follow.Definition 2. 2. 1 S =μ(I, T, A; P) is a Rees matrix semigroup on a cancellative moniod T. A triple (r,σ,Ï€)∈ε(I)×C(T)×ε(A) will be called linked if T isσ- cancellative, whenever either irj orλπμ.Theorem 2. 2. 4 S =μ(I, T, A; P) is a Rees matrix semigroup on a cancellative moniod T. (r,σ,Ï€) is a linked triple of S,thenρ=ρ_r,σ,Ï€ is a good congruence on S.Conversely, for every good congruence of S, there exists a unique linked triple (r,σ,Ï€),such thatρ=ρ(r,σ,Ï€)'In chapter 3, we give a definition of (?)^ρ orthogroup and study the structure of (?)^ρorthogroup. The main results are given in follow.Definition 3. 1. 2 A strong (?)^ρ- abundant semigroup is called a (?)^ρ- orthogroup, if E(S) is a band, and satisfies Ehresmann type condition, that is to say, ET-condition^[9]:Theorem 3. 2. 1 S is a semigroup, the statements below are equivalent:(â…°)There existsρ∈(?)C(S), such that S is a (?)^ρ- orthogroup, and the following condition is satisfied: (C1)xepye, for every e∈E(S).(â…±)S = [Y; S_α= I_α×T_α×A_α], where every S_αis a rectangular unipotent semigroup, and there existsρ_α∈(?)C(T_α), such that T_αisρ_α- left cancellative, E(S) is a band ,and satisfies the following condition:(C2) for any a, b∈T_α,if aραb, then for any (?)β≤α, there exists j∈I_β,μ∈A_β, such thatIn chapter 4, we give a definition of partial Rees matrix semigroup on a cancellative moniod, and the discription of partial Rees matrix semigroup on a cancellative moniod. The main results are given in follow.Theorem 4. 2. 1 A semigroup S has a ideal which is a Rees matrix semigroup on a cancellative moniod (?) there is a isomorphism between S and a M(T; I, A; P; Q, (?), (?),ξ,μ).Theorem 4. 2. 2 There exists a isomorphism between∑= M(T; I, A; P; Q, (?), (?),ξ,μ) and there exists a isomorphismω: T→A→A^*and a isomorphismΩ: Q→Q^*, such that(1)(2)In chapter 5, we give a discription of isomorphism between orthodox super rpp semigroups. The main results are given in follow.Theorem 5. 2. 2 S₁ = are orthodox super rpp semigroups, if there exists isomorphism, such that for (?)αE Y, there exists a isomorphism and a bijective map and any (i,a,λ), thefollowing conditions are satisfied: then the map is a isomorphism.Conversely, every isomorphism from S₁ to S₂ can be so constructed.In chapter 6, we give a definition of left cross product,and study the left cross product of left C-full Ehresmannn semigroups. The main results are given in follow.Theorem 6. 2. 1 T is a semilattice of unipotent semigroup is a semilatrice decomposition of left zero band I_α, then the left cross product I×_φT of I and T is a left C-full Ehresmannn semigroup.Conversely, every left C-full Ehresmannn semigroups can be so constructed. keywords: semigroups, generalized Rees matrix semigroups, (?)^ρ- orthogroups, orthodox super rpp semigroups, partial semigroups, left C-full Ehresmannn semigroups.