Remark On A Semilattice Of Semigroups

In this dissertation, we characterize the translational hull of semilattice of semigroups by the translational hull of semigroups, and study this question on the semilattice of inverse semigroups. We discuss the relations between the kernel normal system and congruences on a semilattice of inverse semigroups S_α and that on every inverse semigroup S_α.The main results are given in follow.In Chapter 1, we give the introduction and preliminaries.In Chapter 2, we introduce the concepts of the congruence keeping consistency after left (right) translation and the congruence keeping consistency after left and right translation. First we characterize the translational hull of semilattice of semigroups by the translational hull of semigroups. Moverover,we give the relations between the translational hull of a semilattice of inverse semigroups and that of these inverse semigroups.The following is the main results.Theorem 2.8 Let S = [Y; S_α, φ_α,β) be a strong semilattice of semigroups S_α.where φ_α,β is a epimorphism,Kerφ_α,β is the congruence keeping consistency after left and right translation of S_α, then Ω(S) is a strong semilattice of Ω(S_α),and Ω(S) = [Y; Ω(S_α), (φ_α,β|-)], where (φ_α,β|-) : (λ,ρ) → ((λ|-),(ρ|-)) ((λ,ρ) ∈ Ω(S_α)), and (λ|-)(aφ_α,β) = (λα)φ_α,β,(aφ_α,β)(ρ|-) = (aρ)φ_α,β (?)_a ∈ S_α.Proposition 2.10 Let semigroup S and semigroup T be simple or 0—simple weakly reductive semigroups, and Ω(S) (?) Ω(T), then S = T.Proposition 2.11 Let S = (Y; S_α) be a semilattice of inverse semigroups S_α, then there exists a monomorphism from Ω(S) to the direct, product of Ω(S_α) (with a zero possibly adjoined).In Chapter 3, First, we discuss the relation between the kernel normal system and the congruence on a strong semilattice of inverse semigroups S_α and that on S_α.Second,we discuss the relation between the kernel normal system and the congruence on a semilatticeof inverse monoids Sa and that on SQ.The following is the main results.Theorem 3.1.5 Let S - [Y; Sa, 4>a,p\ be a strong semilattice of inverse semigroups Sa, {1Ca}aey be a family of admissible kernel normal system of S,thenK = {K= (J Ka\Ka G !Ca,K0 G )C0,KQ^a0 = K^a0^a,(3 6 Y] (1)is a kernel normal system of S, and(a, b) ￡ & <& a ￡ Sa,b ￡ S$, (aa"1)^^, {bblTheorem 3.2.6 Let 5 = (Y, Sa)be a semilattice of inverse monoids So,{lCa}Qey be a family of normal kernel normal system of S,thenK. = {K = J Ka | KQ e Ka,K(, € fy.a < /8,Va G KQ,a ■ lp 6 ^} is a kernel normal system of 5, and 6e = {{a,b) G 5x5|aG 5Q,6G 5^,afl-1-la0,66-1-lQ^!a6-1 G Ka^3Kafi ￡fCa:i}. (2)Further,^ !.??= ^Q-In Chapter 4, we mainly characterize the relations between pmm,Pmax-.p"1"'■ pmnT and {Pmin}aev,{Pmaj:}aev ,{pa'n}aev, {paM}aGV bY the relations between a congruence p and an admissible congruence class {pa}a￡Y on a strong semilattice of inverse semigroup Sa. The following is the main results:Theorem 4.4 Let S = [Y;Sa,atp] be a strong semilattice of inverse semigroup SQ(a G y),and p be a congruence on S induced by admissible congruence class {po}nev-If 4>a,p \Ea: &a —> Ep is a surjection, then pmQx is a congruence on 5 induced by {Pmai}aev; If ^q./S is a injection,and $Q>Jg |^Q: Ea -> E^ is a surjection,then pmm is a congruence on 5 induced by {Pm;nWy-Theorem 4.8 Let S = [Y; Sa,afi IS a bijection,then pmm is a congruence on S induceed by {p?lTl}Q￡Y, and pmax is a congruence on 5 induceed by \Pa JaeY-...