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In this paper, we mainly use the family of semigroups congruence characterize its strong semilattice congruence, and discuss the family of the semigroup congruence lattice lattice direct product with its strong semilattice congruence lattice relationship . In addition to the general semigroup research, the article also made the discussion of issues related to the inverse semigroup, with the formal with strong semilattice. The details are as follows: the first chapter, given in the Introduction and prior knowledge. Chapter II, using mainly family semigroup congruence on the characterization of its strong semilattice congruence and give this family semigroup congruence lattice direct product lattice and its strong semilattice congruence on grid sublattices isomorphic relationship. The quotient semigroup Finally, draw a strong semilattice of semigroups its corresponding semigroup quotient semigroup strong semilattice charge conditions. on α (α ∈ Y) congruence of {ρ α } α ∈ Y strong semilattice S is allowed to define S ρ is: (a, b) ∈ ρ of a ∈ S that α b ∈ S that β (αφ α, αβ-, Bφ β, αβ ∈ ρ αβ , (2.1.2), ρ is a congruence on S and α ∈ Y, ρ | S α = ρ α . the strong semilattice semigroup S = [Y Theorem 2.2.5; the S α ; φ α, β } α ∈ Y ρ, where ρ is {ρ α } α ∈ Y -induced S-congruence. mapping strong semilattice allow for the semigroup S α congruence lattice C to S on congruence lattice L 1 grid isomorphic mapping the third chapter, mainly characterize strong inverse semigroup semilattice congruence congruence and discuss issues related group congruence main conclusions are as follows: Theorem 3.1.6 Let S = [Y; S α ; φ α, β ] congruence pairs of inverse semigroups strong semilattice, (N, τ) for S arbitrary α ∈ Y, Hutchison N ∩ S α < / sub> and τ | E α , respectively, for the N α and τ α . (N α , τ α ) (α ∈ Y) congruence pairs S α Furthermore, if N and τ satisfies: (iii) If α ∈ S α , α ∈ N β ≤ α, the αφ α, β ∈ N (ⅳ) e ∈ E is α f ∈ E β , (e, f) ∈ τ (eφ α, αβ to fφ β αβ ) ∈ τ, (v) {τ | the E α < / sub>} α ∈ Y allow congruence class (N, τ) is just the grounds the (N α the τ α ) (α ∈ Y S-m Sa;) the congruence induced S right. Theorem 3.1.7 gb. flutter strong inverse semigroup semilattice if a. congruence on Sa (Va e Y), (trp ). y satisfies (3.1.2), (KERP} plant meet (3.1.3), and (3.1.4) of formula, then the relationship S on the corpse: h edge Ep,. E where 6 [India - (. pill, where Na, JE within where (3.1.8) is cut * yesterday. induced congruence on S, and Chuan sa-n Mountain In turn, if the corpse is a congruence on S, any aEy, Mountain Srt the congruence on where. Moreover, if p satisfies: Kerp GU Ker (pk) and aey (a, ... EP aE where, bE Ma - Ah 5a Lu, ... pill, 6 rural, JEP, pEg), the corpse just grounds congruence on S P jso induced the fourth chapter, mainly to discuss the minimum half with strong semilattice grid I and informal with a strong semilattice main conclusions are as follows: Theorem 4.2 Let B IY for strong Semilattice with B * ae Y); B Mountains ha is outside s smallest semilattice on Ba congruence is a two Ua: (a, b) a one ga of Xa, b. B, (a, b) a., aEY smallest semilattice congruence on B and B plus BJaJaEY) about the the semilattice Y's strong semilattice. fifth chapter focuses on pure semigroup strong semilattice minimum inverse semigroup congruence of the Theorem 5 stone set S II r SaZ gb playing for the pure semigroup Sa strong semilattice Sa, V EY minimum inverse semigroup congruence relation defined on S the Ah = U Ah, that's bEJ, t [factory ... NE1 - Day EXa bE SaSa / a, NE hh, minimum inverse semigroup S on congruence and v Iso = s. Conversely, if the 7 S minimum inverse semigroup congruence, then W II called Ting ae Y Sa on minimum inverse semigroup congruence and D = UTh.
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