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=[ρT,ρT],ρK=[ρK,ρK],ρV=[ρV,ρV].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(τE,π,τF) bea congruence triple,the relationρ=ρ(τE,π,τF) defined by(i, a,λ)ρ(j, b,μ)(?)(i, aa-1p1i-1, 1)τE(j, bb-1p1j-1, 1), p1iapλ1πp1jbpμ1,(1, pλ1-1a-1a,λ)τF(1, pμ1b-1b,μ).is a congruence on S such thatρE=τE,ρT=π,ρF=τF. Conversely, ifρis a congruenceon S,thenρ, (ρE,ρT,ρF) 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)(?)τE=τ′E,τF=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:ρT=ρ(τE,πt,τF),ρT=ρ(τE,πt,τF),ρV=ρ(VE(π),π, VF(π)),ρV=ρ(VE(π),π, VF(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(?)ρE=θE,ρF=θF;ρKθ(?)ρG=θG. For any congruenceρ,we we find out the minimal and maximal of T-class andK-class::ρT=ρ(τE,ωG,τF),ρT=ρ(τE,τG,τF),ρK=ρ(κE(ξ),ξ,κF(ξ)),ρK=ρ(εE,ξ,εF). |