The Structures And Congruences Of A Distributive Lattice Of Semirings With Additional Zero Elements

The dissertation is divided into three chapters. In Chapter 1, we mainly discuss the structures and congruences of a distributive lattice of semirings with additional zero elements; in Chapter 2, we mainly discuss the structures and congruences of a distributive lattice of inverse semirings with additional zero elements; in Chapter 3, we give two kinds of idempotent semirings and other structures. The results are given in follow.In the first part of Chapter 1, we give the introduction and preliminaries.In the second part of Chapter 1, we mainly give the definition of a distributive lattice of semirings with additional zero elements, and give a characterization of structures and congruences on it. Let D be a distributive lattice, {S_α|α∈D} are a collection of pairwise disjoint semirings with additional zero elements. LetS =∪_α∈DS_α,and S⁺ = ((D, +), (S_α·+)), S^* = ((D,·), (S_α··)). If S satisfies conditions:Then we call S a distributive lattice of semirings with additional zero elements {S_α|α∈D}, written it as S = (D; S_α), and (S, +,·) is a semiring. Main results :Theorem 1.2.3 Let S = (D;S_α), if S satisfies the condition: for every a∈S_α, b∈S_β,(α,β∈D),a relationρon S is defined by Thenρis a semiring congruence on S, and S is a sub direct product of a distributive lattice D and a semiring S/ρ; Conversely, if there exists the same congruence as (1) on S, and 0_α+ 0_β= 0_β+ 0_α(α,β∈D), Then S satisfies (C₃), (C₄).Theorem 1.2.4 Let S = (D;S_α), (?)α,β∈D,0_α+ 0_β= 0_α+β. Then S =＜D;S_α,φ_α,β>.Theorem 1.2.6 Let S = (D;S_α), each S_α((?)α∈D) satisfies the conditions of Lemma 1.2.5. If S satisfies conditions:a relationρon S is defined byThenρis a semiring congruence on S, and (S/ρ, +) is a semilattice. Especially if for everyα∈D, (S_α,+) is commutative, and satisfies the conditionThen S is a subdirect product of an idempotent semiring S⁰ = {0_α|α∈D} and a semiring S/ρ.In the third part of Chapter 1, we give the definition of a family of admissible congruences, and characterize a semiring congruence by way of it.Lemma 1.3.4 Let S = (D;S_α), {ρ_α}_α∈D are a family of admissible congruences on S, a relationρon S is defined by :Thenρis a semiring on S.In the fourth part of Chapter 1, we discuss the relation of a sublattice of the direct product of the lattices of congruences on a family of semirings with additional zero elements and a sublattice of the lattice of congruence on the distributive lattice of those semirings. The main results are given in follow.Theorem 1.4.3 Let S = (D; S_α), (?)δ≥α, S_δ(?) S_α+ 0_δ. Define a mapφ: C→L₁,Π_α∈Dρ_α(?)ρ, whereρis the congruence on S induced by {ρ_α}_α∈D, Thenφis an isomorphism from C, the lattice of admissible congruences on the distributive lattice of semirings with additional zero elements, to L₁.the sublattice of congruences on S.In the last part of Chapter 1, we get a necessary and sufficient condition for a quotient semiring of a distributive lattice of semirings with additional zero elements to be a distributive lattice of quotient semirings with additional zero elements. The main results are given in follow.Theorem 1.5.2 Let S = (D;S_α),σis the corresponding distributive lattice congruenceon S,ρis a congruence on S, for everyα∈D, letρ_α=ρ|_Sα, and suppose that a, b∈S_αand S satisfies the condition :Then S/ρ= (?) is the distributive lattice of semirings with additional zero elements {S_α/ρ_α= (?)}_α∈D, if and only ifρ(?)σ.In the first part of Chapter 2 ,we characterize the relation between congruences and congruence pairs on an inverse semiring. Main conclusions:Theorem 2.1.8 Let S be an inverse semiring, andρis a semiring congruence on S. Then (Kerρ, trρ) is a congruence pair.Conversely, if (N,Ï„) is a congruence pair, then the relation:is a semiring congruence on S. Moreover , Kerρ_(N,Ï„) = N,trρ_(N,Ï„) =Ï„,ρ(Kerρ,trρ) =ρ.In the second part of Chapter 2, we discuss the structures of a distributive lattice of inverse semirings with additional zero elements, that is ,Theorem 2.2.2 Let S = (D;S_α), if S satisfies the condition:a relationρon S is defined byThenρis a semiring congruence on S.and S is a subdirect product of a distributive lattice D and an inverse semiring S/ρ. In the third part of Chapter 2, we mainly discribe a congruence pair on a distributive lattice of inverse semirings with additional zero elements by way of a family of congruence pairs of those semirings. Main results:Theorem 2.3.7 Let S = (D;S_α), and {(N_α,Ï„_α)}_α∈D, a family of (?)-normal congruencepairs of S, (N,Ï„) is defined by :Ï„= {(e,f)∈E⁺(S)×E⁺(S) | e∈E⁺(S_α), f∈E⁺(S_β), (e + 0_α+β,f + 0_α+β)∈τ_α+β}. Then (N,Ï„) is a congruence pair of S, andρ_(N,Ï„) = {(a, b)∈S×S |α∈S_α, b∈S_β, (a' +a + 0_α+β, b' + b + 0_α+β)∈τ_α+β, a + b'∈N_α+β},ρ_N,Ï„ |_Sα=ρ_{(Nα,τα)}.In the last part of Chapter 2, we describe congruences and congruence pairs on a distributive lattice of inverse semirings with additional zero elelments and construct an isomorphism betweem the lattice A, a family of all the (?)-normal congruence pairs on S, and the lattice B, all the (?)- standard congruence pairs on S. The main results are given in follow :Theorem 2.4.4 Let S = (D: S_α), A, a set of all the I-normal congruence pairs of S, and the lattice B, all the (?)- standard congruence pairs of S, we define a relation≤on A and B as follows:Then (A,≤) and(B,≤) are both complete lattices, moreover, A≌B.Corollary 2.4.6 Let S = (D;S_α),p_αis a semiring congruence on S_α,and {ρ_α}_α∈D are a family of (?)-admissible congruences on S. If p is the congruence on S correctly induced by {ρ_α}_α∈D, then Kerρ= (D;Kerρ_α).In Chapter 3, firstly we define a structure of the strong right normal idempotent semiring of V-semirings. That is, when∧is a right normal idempotent semiring, {S_α|α∈D} are a collection of pairwise disjoint V-semirings, where V is a class of semirings, suppose that for eachα∈∧and for each (β∈∧_α∧_α= {γαδα|γ,δ∈∧}∪α∧α∧= {αγαδ|γ,δ∈∧}, there exists a semiring homomorphismφ_α,β : S_α→S_β, satisfying conditions (R₁),(R₂), and define two binary operations on S =∪_{α∈∧}S_αby for any a,b∈S, suppose that a∈S_α, b∈S_β,α,β∈∧.let Then (?) is a semiring, we call it a strong right normal idempotent semiring of V-semirings. And by this we have the structures of the normal Type A-idempotent semiring which arises as a right normal idempotent semiring of left zero idempotent semirings, and some corollaries. Secondly, we give the definition of the pseudo-strong right normal idempotent semiring of V-semirings. And we have the additive normal Type B-idempotent semiring which arises as a pseudo-strong right normal idempotent semiring of left zero semirings. And we also have the additive normal Type B-idempotent semiring which arises as a pseudo-strong semilattice idempotent semiring of rectangular semirings, and some corollaries.Theorem 3.2.4 A semiring S is a normal Type A-idempotent semiring, if and only if S is a strong right normal idempotent semiring of left zero idempotent semirings.Corollary 3.2.5 A semiring S is a [ left normal, rectangular, left zero ] Type A-idempotent semiring, if and only if S is a strong [ semilattice, right zero, trivial ] idempotent semiring of left zero idempotent semirings.Theorem 3.2.8 S is a direct product of a normal Type A-idempotent semiring ans a band ring with an identity 1, if and only if S is a strong right normal idempotent semiring of Type A-left rings.Theorem 3.3.6 S is an Type B-idempotent semiring, then S is an additive normal idempotent semiring, if and only if S is a pseudo-strong semilattice idempotent semiring of rectangular semirings.Theorem 3.3.9 A semiring S is an additive normal Type B-idempotent semiring, if and only if S is a pseudo-strong right normal idempotent semiring of left zero semirings.