On Structures And Congruences Of Some Semirings

The dissertation is divided into three chapters. In Chapter 1,we give the intro-duction and preliminaries；in Chapter 2,we mainly discuss additive regular kernel normal system of an orthodox semiring;in Chapter 3,we give the structures and congruences of a distributive lattice of semirings with additional zero elements.The main results are given in follow.In Chapter 1,we give the introduction and preliminaries.In the first part of Chapter 2,we give the preliminaries.In the second part of Chapter 2,we give a definition of additive regular kernel normal system of an orthodox semiring,and we have additive regular kernel normal system and congruence which are one to one.The main results are given in follow.Define 2.2.1 The set B={Bi:i∈I} is defined to be an additive regular kernel normal system of an orthodox semiring S if(K1)each Bi is a regular subsemiring of S；(K2)Bi∩Bj=(?),if i≠j；(K3)each additive idempotent of S is contained in some Bi；(K4)for each a∈S,a'∈V+(a),andi∈I,there is some j=j(a,a',i)∈I,such that a'+Bi+a(?)Bj；(K5)for each i,j∈I,there is some k∈I such that Bi+Bj+Bi(?)Bk；(K6)if a,a+b,b+b',b'+b∈Bi for someb'∈V+(b),then b∈Bi；(K7)for each i∈I and for each j∈I,there is some k∈I such that Ei+Ej(?)Ek,where Ei is the set of additive idempotents of Bi；(K8)for each x∈S,for each Bi∈B,there is some j,k∈I such that Bix (?) Bj,xBi(?)Bk.Theorem 2.2.8 Ifρis a congruence on an orthodox semiring S,then the additive regular kernel B ofρis an additive regular kerncl normal system of S,andρ=tρB,the additive transitive closure of the relation ofρB defined by(5)； Conversely,if B is an additive regular kernel normal system of S,then there is pre-cisely one congruence p on S such that B is the additive regular kernel ofρandρ=tρB.In the third part of Chapter 2, we characterize a congruence which seperates the additive idempotents by additive regular kernel normal system, and determine a necessary and sufficient condition for Green's equivalence H+ to be a congruence on an orthodox semiring. The main results are given in follow.Define 2.3.3 Let S be an orthodox semiring,we define a set N={Ne:e∈E+(S)} of normal subgroup of the maximal subgroups{He:e∈E+(S)} of S to be an additive group kernel normal system of S,if the Ne satisfy the conditions:(i) a'+Ne+a(?)Na'+e+a (?)a∈S,a'∈V+(a),e∈E+(S).(ii) Ne+Nf(?)Ne+f (?)e,f∈E+(S).Theorem 2.3.5 If p is an additive idempotent-separating congruence on an or-thodox semiring S,then the kernel N ofρis an additive group kernel normal system of S,andρ=ρN,the relation defined by (8);Conversely,if N is an additive group kernel normal system of S,then there is pre-cisely one congruence p on S such that N is the kernel of p,this congruence p is an additive idempotent-separating congruence on S andρ=ρN.Theorem 2.3.6 A necessary and sufficient condition for H+ to be a congruence on an orthodox semiring S is that the set{He:e∈E+(S)} of maximal subgroups of S satisfy the condition He+Hf(?)He+f (?)e,f∈E+(S).In the first part of Chapter 3, we give the preliminaries.In the second part of Chapter 3,we mainly give the definition of a quasi distributive lattice of semirings with additional zero elements, and give a characterization of structures and congruences on it.The main results are given in follow.Define 3.2.1 Let D be a distributive lattice,{Sα|α∈D} are a collection of pairwise disjoint semirings with additional zero elements.LetS=∪α∈DSα,and S+= ((D,+), (Sα.+)), S*=((D,·), (Sα.·)). If S satisfies conditions: Then we call S a quasi distributive lattice of semirings with additional zero elements {Sα|α∈D}, written it as S=[D; Sα].Theorem 3.2.3 Let S= [D;Sα], if S satisfies the condition:for every a∈Sα, b∈Sβ,(α,β∈D), a relation p on S is defined by Then p is a semiring congruence on S, and S is a subdirect product of a distributive lattice D and a semiring S/p; Conversely, if there exists the same congruence as (1) on S, and 0α·0β=0β·0α(α,β∈D), Then S satisfies (C4), (C5).Theorem 3.2.4 LetS=[D;Sα],(?)α,β∈D,0α·0β=0αβ. Then S=(D;Sα,ψα,β).In the third part of Chapter 3, we give the definition of a family of quasi admissible congruences, and characterize a semiring congruence by way of it.The main results are given in follow.Theorem 3.3.4 Let S=[D;Sα],{ρα}α∈D are a family of quasi admissible con-gruences on S, a relation p on S is defined by: Then p is a semiring congruence on S.In the fourth part of Chapter 3, 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 quasi distributive lattice of those semirings. The main results are given in follow.Theorem 3.4.4 Let S=[D;Sα],(?)δ≤α, Sδ(?)Sα·0δ.Define a map (?):C→L1,Πα∈Dρa(?)ρ,whereρis a congruence on S induced by{ρα}α∈D,then (?) is a lattice isomorphism.In the last part of Chapter 3, we get a necessary and sufficient condition for a quotient semiring of a quasi distributive lattice of semirings with additional zero elements to be a quasi distributive lattice of quotient semirings with additional zero elements. The main results are given in follow.Theorem 3.5.2 Let S=[D;Sα],σis the corresponding distributive lattice con-gruence on S,ρis a congruence on S.(?)α∈D,letρα=ρ|Sα, and suppose that a.b∈Sa and S satisfies the condition: then S/ρ=S is the quasi distributive lattice of semirings with additional zero elements {Sα/ρα=Sα}α∈D(?)ρ(?)σ.