Some Semi-ring Structure

In this dissertation, we characterize the congruence on a strongly distributive lattice of semirings by the congruence on those semirings,and give a characterization of constructureof pseudo-strongly distributive lattice of bisemirings;besides,we mainly give some distributive congruences and ring. congruences,we discuss the regular semirings whose additive reduct are special semigroups.The main results are given in follow:In the first chapter , we give the introductions and preliminaries.In the second chapter , we characterize the congruence on a strongly distributive lattice of semirings.by the congruence on those semirings,The main results are given in follow:Theorem 2.2.1 Let S =＜D;S_α,φ_α,β＞,ρ_αis the the semirings congruence on S_α,anid {ρ_α|α∈D} satisfy condition: .. .a relationρon S is defined by:Thenρis a congruence on S.Theorem 2.2.3 Let S =< D; S_α,φ_α,β >.σis the corresponding distributive lattice congruence on S,ρis a congruence on S, (?)α∈D, letρ_α=ρ|s_α,and the following condition is satisfied,i.e.Then S/ρ= (?) is the strongly distributive lattice of S_α/ρ_α= (?) if and only if (?).In the third chapter ,we give a definition of pseudo-strongly distributive lattice of bisemirings;besides, we give a characterization of constructure which is subdirect of a distributive lattice and S/θ.The main results are given in follow.Define 3.1.1 Let (D,+,·) be a distributive lattice,{(?),α∈D} be a family of pairwise disjoint besemirings indexed by D, For each pair a≥βin D,there exists a monomorphism of bisemiringsφ_α,β:(?). such that:On (?) the operations are denned as follows:(?)and c satisfiesThe system is called pseudo-strongly distributive lattice of bisemiring varieties.We write.Theorem 3.1.3 Let S =< D; (?),φ_α,β >,define.θon S:Thenθis a congruence on S.Theorem3.1.6 Let S =< D;(?),φ_α,β >,θis defined in.Theorem 3.1.3, Then S is pseudo-subdirect product of a D and S/θ.In Chapter 4 ,we mainly give some distributive congruences and ring congruences.The main results are given in follow.:Lemma 4.1.1 Let S is a semiring,(?) e∈E⁺ (S),defineηon S:Thenηis a semiring congruence on S,and (S/η, +) is a semilattice.Lemma 4.1.2 Let S is a distributive semiring,(?) e∈E(S),defineρon S/η: Thenρis a congruence on S/η.and ((S/η)/ρ,·) is a semilattice.Theorem 4.1.3 Let S is a distributive semiring,(?) a,b∈S, defineθon S:Thenθis a distributive congruence on S .Theorem 4.2.2 Let S is a commute semirings whose additive reduct is inverse semigroup,defineρon S:Thenρis a +-semilattice congruence on S.Theorem 4.3.3 Let S is a semiring which reduct is left normal band,define D⁺ on S:...