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This paper is divided into three chapters. Chapter Introduction section introduces regular semigroups, completely simple semigroup congruence scored and inverse semigroups Rees matrix semigroup description of some congruence. First regular semigroup S, the congruence on (k, τ) is uniquely determined by the congruence form the ρ (K, τ) : the αρ (K, τ) < / sub> b (?) a (LτLτL ∩ RτRτR) b, ab '∈ K ((?) b' ∈ V (b)). Congruence lattice C (S), where each class T congruence. Provide constant, K class and V class are interval, ie ρT = [the ρ T the ρ T Completely simple semigroup can be expressed for the group G Rees matrix semigroup, Hutchison S = M (I, G, ^; P). I ∧ equivalence relations and group G on congruence normal subgroup constitutes allowable triples (γ, N, π) is uniquely determined. And For inverse semigroups matrix semigroup S = M (I, T, ^; P), the analog completely simple semigroup broader semigroup. Some described the same I may be allowed to triples (φ, π, ψ), but this approach can not be completely ruled it congruence. The second chapter studies the inverse semigroups Rees matrix semigroup congruence lattice. This section gives the inverse semigroups Rees matrix semigroup S = M (I, T, ^; P) defined congruence scored three important sub-semigroup E, F, A, respectively with E, F, abstract congruence on T scored congruence triples. Set (τ E , π, τ F ) is congruence triples, the relationship p = p (τ E π, τ F ) : (i, a, λ) ρ (j, b, μ) (?) (i, aa -1 p < sub> 1i -1 , 1) τ E (j, bb -1 p 1j -1 , 1), p 1i ap λ1 πp 1j bp μ1 , (1, p λ1 -1 a -1 a, λ) τ F (1, p μ1 b -1 b, μ). S congruence, and ρ E = τ E , ρ T = π, ρ F = τ F . In turn, set p is any congruence of S (the ρ E , the ρ T the ρ F ) is a congruence triple and ρ = ρ (ρ E , ρ T , ρ F ) . As a result, we are given a more natural equivalence relation on the Congruence Lattice T, V characterization: ρ (τ E , π, τ F ) T ρ (τ ' E , π', τ ' F ) (? ) τ E = τ ' E , τ F = τ' F , ρ (τ < sub> E , π, τ F ) V ρ (τ ' E , π' , τ ' F ) (?) π = π'. For any congruence ρ, we have identified a great class T and V class, minimal congruence: ρT = ρ (τ , E the pi t , τ < sub> F ), ρ T = ρ (τ E , π t , τ F < / sub>) , ρ V = ρ (V E (π), π, V F (π )) , ρ V = ρ (V E (π), π, V F (π)) . The third chapter is an application of the results of the second chapter, gives a completely simple semigroup congruence on nuclear track method. And then gives the the Congruence Lattice equivalence relation T, K the simple characterization: ρTθ (?) The ρ E = the theta E, the ρ F < / sub> = θ F ; ρKθ (?) ρ G = θ G . For any congruence ρ, we get: the ρ T the = p (the τ E the omega G , the τ F < / sub>) , ρ T = ρ (τ E , τ G , τ F < / sub>) , ρ K = ρ (κ E (ξ), ξ, κ F (ξ )) , ρ K = ρ (ε E , ξ, ε F ) .
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