Congruences And Structures Of Semigroups

In this dissertation.we mainly study the congruences onÏ€-regular semigroups with the kernel and the trace of congruences. Futher we apply our results to GV-semigroups and E-inversive semigroups. There are five chapters:In the first chapter , we give the introductions and preliminaries.In the second chapter ,we give the definition of the normal subset of aÏ€-regular semigroup S , the normal equivalence on E(S) and then we give the description of completely regular congruence pairs of S. The main results are given in follow:We represent the set of completely regular congruences on aÏ€-regular semigroup S by CRC(S) and the set of completely regular congruence pairs on aÏ€-regular semigroup S by CRCP(S).Definition 2.1.4 Let (K,ξ)∈CRCP(S) and define a binary relationρ_(K,ξ) on S bywhere r(a) = a^m,r(b) = bⁿ.Theorem 2.1.15 Let (K,ξ)∈CRCP(S).Thenρ_(K,ξ)∈CRC(S) and Kerρ_(K,ξ) =K,trρ_(K,ξ)=ξ.Conversely,ifρ∈CRC(S), then (Kerρ,trρ)∈CR.CP(S) andρ=ρ_{(Kerρ,trρ)}.Theorem 2.1.17 The mappings defined byare order preserving complete lattice isomorphisms and (?)₁ =(?)₂^-1We represent the set of completely regular congruences on a GV-semigroup S by GVCRC(S) and the set of completely regular congruence pairs on a GV-semigroup S by GVCRCP(S). Definition 2.7.3 Let (K,ξ)∈GVCRCP(S) and define a binary relationρ_(K,ξ) on S bywhere r(a) = a^m.Theorem 2.7.4 Let (K,ξ)∈GVC RCP(S).Thenρ_(K,ξ)∈GVCRC(S) and Kerρ_(K,ξ)=K,trρ_(K,ξ)=ξ.Conversely,ifρ∈GVCRC(S), then (Kerρ, trρ)∈GVCRCP(S) andρ=ρ_{(Kerρ,trρ)}.Theorem 2.7.5 The mappings defined byare order preserving complete lattice isomorphisms and (?) ₁=(?)₂^-1.In the third chapter ,we mainly study the relation between a congruence on a GVsemigroupS and its trace on S_α.Theorem 3.2.1 Letρbe a congruence on S and S/ρbe a semilattice of nilextensionsof left groups. Then for arbitraryα∈Y,S_α/ρ_αis a nil-extension of a left group,whereρ_α=ρ| S_α.Theorem 3.2.6 Letρ_αbe a congruence on S_αfor arbitraryα∈Y and S_α/ρ_αis a nil-extension of a left group. Thenρ= (∪_ρα )^# is a congruence on S and S/ρis a semilattice of nil-extensions of left groups.Theorem 3.3.10 Let v = {(e, f)∈E(S)×E(S) | eR^*f}^#. Then v is the smallest congruence on S such that S/v is a semilattice of nil-extensions of left groups.In chapter four,we give the definition of the normal subsemigroup of a semigroup S, the normal congruence on < E(S) > and then we give the description of rectangular group congruence pairs on S. The main results are given in follow:We represent the set of rectangular group congruences on an E-inversive semigroup S by ERCP(S) and the set of rectangular group congruence pairs on an E-inversive semigroup S by ERC(S).Definition 4.2.5 Let (K,ξ)∈ERCP(S) and define a binary relationρ(K,ξ)on S by Theorem 4.2.6 Let (K,ξ)∈ERCP(S). Thenρ_(K,ξ)∈ERC(S) and Kerρ_(K,ξ)= K,htrρ_(K,ξ) =ξ. Conversely,ifρ∈ERC(S), then (Kerρ,htrρ)∈ERCP(S) andρ=ρ_{(Kerρ,htrρ)}.Theorem 4.2.7 The mappings defined byare order preserving complete lattice isomorphisms and (?)₁ =(?)₂^-1.In chapter five.we study the structure of B^*-pure semigroups by denning a partial order.Theorem 5.2.5 The following conditions are equivalent:(1)S is a GV-inverse semigroup and S² (?) RegS ;(2)S is a B^*-pure semigroup;(3)S is a strong semilattice ofÏ€-groups and S² (?) RegS.