Of Inverse Semigroups, Rees Matrix Semigroup Congruence Lattice

The paper has three parts. In chapter one,characterizations of congruences on regularsemigroups and completely simple semigroups are introduced.The description of some con-gruences on Rees matrix semigroups over an inverse semigroup is also presented.Any con-gruence on a regular semigroup S is uniquely determined by its associated congruence pair(K,τ) asρ_(K,τ) such that:αρ_(K,τ)b(?)a(LτLτL∩RτRτR)b, ab′∈K ((?)b′∈V(b)).It is proved that any T- class,K- class and V- class are respectively intervals,thatis for any congruenceρ,ρT=[ρ_T,ρ^T],ρK=[ρ_K,ρ^K],ρV=[ρ_V,ρ^V].Congruences on acompletely simple semigroup S=M(I, G,∧; P) are uniquely determined by means ofadmissible triples (γ, N,π) whereγandπare equivalences of I and∧respectively andN is a normal subgroup of G.Rees matrix semigroup over an inverse semigroup whichis demoted by S=M(I, T,∧; P) is a generalization of completely simple semigroups.Some congruences on it are described by congruence triplesφ,ψ,πwhereφandψareequivalences of I and∧respectively andπis a congruence on T,but not all the congru-ences can be characterized in this way.Chapter two,it is studied the congruence lattice of Rees matrix semigroups over an in-verse semigroup.The congruences on it are characterized.For S=M(I, T,∧; P),threeimportant sub-semigroups E, F, A of it are defined respectively,then congruences triplesof S are abstractly characterized by congruences of E, F, A respectively,Let(τ_E,π,τ_F) bea congruence triple,the relationρ=ρ_{(τE,π,τF)} defined by(i, a,λ)ρ(j, b,μ)(?)(i, aa^-1p_1i^-1, 1)τ_E(j, bb^-1p_1j^-1, 1), p_1iap_λ1πp_1jbp_μ1,(1, p_λ1^-1a^-1a,λ)τ_F(1, p_μ1b^-1b,μ).is a congruence on S such thatρ_E=τ_E,ρ_T=π,ρ_F=τ_F. Conversely, ifρis a congruenceon S,thenρ, (ρ_E,ρ_T,ρ_F) is a congruence triple for S,andρ=ρ_{(ρE,ρT,ρF)}.Hence we cancharacterize the equivalences T, V on C(S) naturally:ρ_{(τE,π,τF)}Tρ_{(τ′E,π′,τ′F)}(?)τ_E=τ′_E,τ_F=τ′_F,ρ_{(τE,π,τF)}Vρ_{(τ′E,π′,τ′F)}(?)π=π′. For any congruenceρ,we find out theminimal and maximal of T-class and V-class:ρ_T=ρ_{(τE,πt,τF)},ρ^T=ρ_{(τE,π^t,τF)},ρ_V=ρ_{(VE(π),π, VF(π))},ρ^V=ρ_{(V^E(π),π, V^F(π))}.Chapter three is an application of the results of the chapter two,we give the kernel-trace approach to congruences on completely simple semigroups and characterize theequivalences T, K on C(S) in a simple way:ρTθ(?)ρ_E=θ_E,ρ_F=θ_F;ρKθ(?)ρ_G=θ_G. For any congruenceρ,we we find out the minimal and maximal of T-class andK-class::ρ^T=ρ_{(τE,ωG,τF)},ρ_T=ρ_{(τE,τG,τF)},ρ^K=ρ_{(κE(ξ),ξ,κF(ξ))},ρ_K=ρ_{(εE,ξ,εF)}.