The Structure And Congruence Of Some Bi-Semirings And Semirings

In this dissertation,we mainly discuss the structures and congruences of the bi-semirings and semirings,we mainly discuss the structures and congruences of a quasi distributive lattice of inverse semirings with multiplication identical elements and a dis-tributive lattice of inverse bi-semirings with additional zero elements. The dissertation is divided into four chapters:In the first chapter,we give the introductions and preliminaries.In the second chapter,we mainly discuss the congruences of a quasi distributive lattice of semirings with multiplication identity elements.The main results are given as follows:Theorem2.1.4Let S=[D; Sα],{ρα}α∈D are a family of admissible congruences on S. A relation p on S is defined by: apb, a∈Sα,b∈Sβ(?)(a·1αβ, b·1αβ)∈ραβ,,Then p is a semiring congruence on S.Theorem2.1.6Let {ρα}α∈D are a family of admissible congruences on S=[D; Sα], if p is the congruence on S correctly induced by {ρα}α∈D. Then ρα=ρ|Sα (α∈D).Lemma2.2.1Let S=[D; Sα], and (?)δ≤a, Sδ(?)Sα·1δ, then Cis the sublattice of ΠLα.Lemma2.2.2Let ρ∈L1, then {ρ|Sα}α∈Dare a family of admissible congruences onS=[D;iSα]and ρ is the congruence on S correctly induced by{ρ|Sα}α∈D.Theorem2.2.3Let S=[D; Sα], and (?)δ<α,Sδ(?)Sα·1δ. Define a map ψ: C→L1,Πα∈Dρα(?)ρ,where ρ is the congruence on S correctly induced by{ρα}α∈D, then ψ is an isomorphism from C. Theorem2.3.1Let S=[D;Sα],σ is the corresponding distributive lattice con-gruence on S, p is congruence on S, Va G D, let ρα=ρ|Sα, if (?)a,b∈Sα (?)δ≤α,(a?1δ, b·1δ)∈ρδ(?)(a,b)∈ρα. Then S/p=S is the quasi distributive lattice of semirings with multiplication identical elements {Sα/ρα=Sα}α∈D if and only if(?)ρ(?)σ.In the third chapter,we mainly discuss the structures and congruences of a quasi distributive lattice of inverse semirings with multiplication identical elements.The main results are given as follows:Lemma3.1.3In defining1.2.6, Va G D, if Sα be an inverse semiring with multipli-cation identical elements, then S=[D; Sα]be an inverse semiringTheorem3.1.4Let S=[D; Sα], if S satisfies the condition:(?)α∈Sα, b∈Sβ, α,β∈D, α·1αβ=b·1αβ(?)δ≤αβ, a·1δ=b·1δ. a relation p on S is defined by: aρb, a∈Sα, b∈Sβ, α,β∈D(?)a·1αβ=b·1αβ,Then p is a semiring congruence on S, and S is a subdirect product of a distributive lattice D and an inverse semiring S/p.Theorem3.2.7Let S=[D; Sα],and {(Nα, τα)}α∈Dis a family of I-normal con-gruence pairs of S.(N,τ)is defined by.N=∪α∈DNα,τ={(e,f)∈E’{S)×E’{S)|e∈E’(Sα), f∈E’(Sβ),(e·1αβ, f·1αβ)∈ταβ}, then (N, r)is a congruence pair of S.In the fourth chapter,we mainly discuss the structures and congruences of a dis-tributive lattice of inverse bi-semirings with additional zero elements.The main results are given as follows:Theorem4.1.2In defining1.2.10,(?)a G D, Sα be an inverse bi-semirings with additional zero elements, then S=(D,Sα)be an inverse bi-semirings.Theorem4.2.6Let S=[D; Sα],and {(Nα,τα)}α∈Dis a family of I-normal con-gruence pairs of S.(N,τ)is defined by.N=∪Uα∈DNα,τ={(e,f)∈E+(S)×E+(S)|e∈ E+(Sα), f∈E+(Sβ),(e+0α+β,f+0α+β)∈τα+β}, then (N,τ)is a congruence pair of S and ρ(N, τ)={(a, b)∈S×S|α∈Sα, b∈Sβ (α’+α+0α+β,b’+b+0αβ)∈τα+β,a+b’∈Nα+β},ρ(N,τ)|Sα=ρ(Nα,τα).