加法含零半环的分配格的结构和同余
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摘要
本文主要分为三大章节,第一章节讨论的是加法含零半环的分配格的结构和同余;第二章节讨论的是加法含零逆半环的分配格的结构和同余;第三章节给出两类幂等半环,并刻划了它们的结构及一系列相关问题.具体内容如下:
第一章:第一节,给出引言和预备知识.
第二节,给出加法含零半环的分配格的定义并刻划了它的结构和同余.设D为分配格,{S_α|α∈D].为一族两两非交的加法含零半环,令S=∪_(α∈D)S_α,且S~+=((D,+),(S_α,+)),S~*=((D,·),(S_α,·)),若满足:则称S为加法含零半环{S_α|α∈D}的分配格,记为S=(D;S_α),且(S,+,·)为半环.主要结论如下:
定理1.2.3设S=(D;S_α),若满足:(?)∈s_α,b∈S_β,(α,β,∈D),在S上定义关系ρ:则ρ为S上的半环同余,且S为分配格D和半环S/ρ的次直积;反之,若S=(D,S_α)上存在形如(1)式定义的半环同余p,且0_α+0_β=0_β+.0_α,(α,β∈D),则S满足(C_3),(C_4).
定理1.2.4设S=(D;S_α),若(?)α,β∈D,0_α+0_β卢=0_(α+β),则S=(D;S_α,φ_(α,β)).
定理1.2.6设S=(D;S_α),其中S_α((?)∈D)满足引理1.2.5中的条件.若S满足:在S上定义关ρ:则ρ为S上的半环同余,且(S/ρ,+)为半格.特别若(S_α,+)((?)α∈D)可交换,则S为幂等半环S~0={0_α|α∈D)和半环S/ρ的次直积.
第三节,引入了容许同余族的概念,并利用容许同余族得到了S=(D;S_α)上的半环同余,即
引理1.3.4设S=(D;S_α),{ρ_α}_(α∈D)为S上的容许同余族,在S上定义关系ρ如下:则ρ为S上的半环同余.
第四节,构造每个加法含零半环同余格的直积的子格与其分配格上的同余格的子格的同构关系.主要结论如下:
定理1.4.3设S=(D;S_α),且(?)δ≥α,S_δ(?)S_α+0_δ.定义映射φ:C→L_1,Π_(α∈D)ρ_α(?)ρ,其中ρ恰为由{ρ_α}_(α∈D)诱导生成的同余,则φ为格同构.
第五节,得出加法含零半环的分配格的商半环为其相对应的加法含零半环的商半环的分配格的充要条件.主要结论:
定理1.5.2设S=(D;S_α),σ称为S上的分配格同余,ρ为S上的同余,(?)α∈D,令ρ_α=ρ|S_α,满足(?) a,b∈S_α,则S/ρ=(?)为加法含零半环{S_α/ρ_α=(?))_(α∈D)的分配格(?).
第二章:第一节,刻划了逆半环上的同余与同余对的关系.主要结论如下:
定理2.1.8设S为逆半环,ρ为S上的半环同余,则(Kerp,trp)为S上的同余对.反之,若(N,τ)为S上的同余对,则关系为S上的半环同余,且Kerρ_((N,τ))=N,trρ_((N,τ))=τ,ρ(Kerρ,trρ)=ρ.
第二节,刻划了加法含零逆半环的分配格的结构.主要结论如下:
定理2.2.2设S=(D;S_α),满足:(?)α∈S_α,b∈S_β,α,β∈D,定义S上的关系ρ:则ρ为S上的半环同余,且S为分配格D和逆半环S/ρ的次直积.
第三节,引入了I-容许同余族,I-标准同余对,I-正规同余对族的概念.利用一族加法含零逆半环上的同余对刻划了其分配格上的同余对.主要结论如下:
定理2.3.7设S=(D;S_α),{(N_α,τ_α)}_(α∈D)为S上的I-正规同余对族,令则(N,τ)为S上的同余对且ρ(N,τ)={(a,b)∈S×s | a∈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,τ)|_(S_α)=ρ_((N_α,τ_α)).
第四节,首先讨论了S上的I-正规同余对族与I-标准同余对之间的关系及一系列相关问题,得出S上所有的I-正规同余对族与I-标准同余对为同构关系,最后的出S上的I-容许同余族诱导生成的同余ρ满足Kerρ=(D;Kerρ_α).主要结论如下:
定理2.4.4设S=(D;S_α),A={S的I-正规同余对族},B={S的所有I-标准同余对}.在A,B上定义关系:则(A,≤),(B,≤)均为完备格,且A≌B.
推论2.4.6设S=(D;S_α),ρ_α为S_α上的半环同余且{ρ_α)_(α∈D)为S上的I-容许同余族,如果ρ为由{ρ_α}_(α∈D)诱导生成的同余,则Kerρ=(D;Kerρ_α).
第三章,首先构造V-半环的强右正规幂等半环的结构,即令∧为右正规幂等半环,{S_α|α∈∧)为一族两两非交的V-半环,V表示半环类.(?)∈∧,(?)∈∧α∧α={γαδα|γ,δ∈∧)∪α∧α∧={αγαδ|γ,δ∈∧),均存在S_α到S_β的半环同态φ_(α,β),即φ_(α,β):S_α→S_β,满足(R_1),(R_2),在集合S=∪_(α∈∧)S_α定义二元运算(?)(?) a,b∈S,设a∈S_α,b∈S_β,α,β∈∧,则(?)为半环,称之为V-半环的强右正规幂等半环.利用这一结构证明了正规的型A-幂等半环是左零幂等半环的强右正规幂等半环及相关推论.与它平行地构造了V-半环的伪强右正规幂等半环,由这一结构证明了加法正规的型B-幂等半环为矩形半环的伪强半格幂等半环,也为左零半环的伪强右正规幂等半环,及相关推论.主要结论如下:
定理3.2.4半环S为正规的型A-幂等半环当且仅当S为左零幂等半环的强右正规幂等半环.
推论3.2.5半环S为[左正规,矩形,左零]型A-幂等半环当且仅当S为左零幂等半环的强[半格,右零,平凡]幂等半环.
定理3.2.8 S为正规型A-幂等半环和含幺带环的直积当且仅当S为型A-左环的强右正规幂等半环.
定理3.3.6设S为型B-幂等半环,则S为加法正规幂等半环当且仅当S是矩形半环的伪强半格幂等半环.
定理3.3.9 S为加法正规的型B-幂等半环当且仅当S为左零半环的伪强右正规幂等半环.
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∈D)S_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 (C_3), (C_4).
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 S~0 = {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→L_1,Π_(α∈D)ρ_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 L_1.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|_(S_α), 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,τ) is a congruence pair, then the relation:is a semiring congruence on S. Moreover , Kerρ_((N,τ)) = N,trρ_((N,τ)) =τ,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,τ_α)}_(α∈D), a family of (?)-normal congruencepairs of S, (N,τ) is defined by :
τ= {(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,τ) is a congruence pair of S, andρ_((N,τ)) = {(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,τ) |_(S_α)=ρ_((N_α,τ_α)).
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 (R_1),(R_2), 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.
第一章:第一节,给出引言和预备知识.
第二节,给出加法含零半环的分配格的定义并刻划了它的结构和同余.设D为分配格,{S_α|α∈D].为一族两两非交的加法含零半环,令S=∪_(α∈D)S_α,且S~+=((D,+),(S_α,+)),S~*=((D,·),(S_α,·)),若满足:则称S为加法含零半环{S_α|α∈D}的分配格,记为S=(D;S_α),且(S,+,·)为半环.主要结论如下:
定理1.2.3设S=(D;S_α),若满足:(?)∈s_α,b∈S_β,(α,β,∈D),在S上定义关系ρ:则ρ为S上的半环同余,且S为分配格D和半环S/ρ的次直积;反之,若S=(D,S_α)上存在形如(1)式定义的半环同余p,且0_α+0_β=0_β+.0_α,(α,β∈D),则S满足(C_3),(C_4).
定理1.2.4设S=(D;S_α),若(?)α,β∈D,0_α+0_β卢=0_(α+β),则S=(D;S_α,φ_(α,β)).
定理1.2.6设S=(D;S_α),其中S_α((?)∈D)满足引理1.2.5中的条件.若S满足:在S上定义关ρ:则ρ为S上的半环同余,且(S/ρ,+)为半格.特别若(S_α,+)((?)α∈D)可交换,则S为幂等半环S~0={0_α|α∈D)和半环S/ρ的次直积.
第三节,引入了容许同余族的概念,并利用容许同余族得到了S=(D;S_α)上的半环同余,即
引理1.3.4设S=(D;S_α),{ρ_α}_(α∈D)为S上的容许同余族,在S上定义关系ρ如下:则ρ为S上的半环同余.
第四节,构造每个加法含零半环同余格的直积的子格与其分配格上的同余格的子格的同构关系.主要结论如下:
定理1.4.3设S=(D;S_α),且(?)δ≥α,S_δ(?)S_α+0_δ.定义映射φ:C→L_1,Π_(α∈D)ρ_α(?)ρ,其中ρ恰为由{ρ_α}_(α∈D)诱导生成的同余,则φ为格同构.
第五节,得出加法含零半环的分配格的商半环为其相对应的加法含零半环的商半环的分配格的充要条件.主要结论:
定理1.5.2设S=(D;S_α),σ称为S上的分配格同余,ρ为S上的同余,(?)α∈D,令ρ_α=ρ|S_α,满足(?) a,b∈S_α,则S/ρ=(?)为加法含零半环{S_α/ρ_α=(?))_(α∈D)的分配格(?).
第二章:第一节,刻划了逆半环上的同余与同余对的关系.主要结论如下:
定理2.1.8设S为逆半环,ρ为S上的半环同余,则(Kerp,trp)为S上的同余对.反之,若(N,τ)为S上的同余对,则关系为S上的半环同余,且Kerρ_((N,τ))=N,trρ_((N,τ))=τ,ρ(Kerρ,trρ)=ρ.
第二节,刻划了加法含零逆半环的分配格的结构.主要结论如下:
定理2.2.2设S=(D;S_α),满足:(?)α∈S_α,b∈S_β,α,β∈D,定义S上的关系ρ:则ρ为S上的半环同余,且S为分配格D和逆半环S/ρ的次直积.
第三节,引入了I-容许同余族,I-标准同余对,I-正规同余对族的概念.利用一族加法含零逆半环上的同余对刻划了其分配格上的同余对.主要结论如下:
定理2.3.7设S=(D;S_α),{(N_α,τ_α)}_(α∈D)为S上的I-正规同余对族,令则(N,τ)为S上的同余对且ρ(N,τ)={(a,b)∈S×s | a∈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,τ)|_(S_α)=ρ_((N_α,τ_α)).
第四节,首先讨论了S上的I-正规同余对族与I-标准同余对之间的关系及一系列相关问题,得出S上所有的I-正规同余对族与I-标准同余对为同构关系,最后的出S上的I-容许同余族诱导生成的同余ρ满足Kerρ=(D;Kerρ_α).主要结论如下:
定理2.4.4设S=(D;S_α),A={S的I-正规同余对族},B={S的所有I-标准同余对}.在A,B上定义关系:则(A,≤),(B,≤)均为完备格,且A≌B.
推论2.4.6设S=(D;S_α),ρ_α为S_α上的半环同余且{ρ_α)_(α∈D)为S上的I-容许同余族,如果ρ为由{ρ_α}_(α∈D)诱导生成的同余,则Kerρ=(D;Kerρ_α).
第三章,首先构造V-半环的强右正规幂等半环的结构,即令∧为右正规幂等半环,{S_α|α∈∧)为一族两两非交的V-半环,V表示半环类.(?)∈∧,(?)∈∧α∧α={γαδα|γ,δ∈∧)∪α∧α∧={αγαδ|γ,δ∈∧),均存在S_α到S_β的半环同态φ_(α,β),即φ_(α,β):S_α→S_β,满足(R_1),(R_2),在集合S=∪_(α∈∧)S_α定义二元运算(?)(?) a,b∈S,设a∈S_α,b∈S_β,α,β∈∧,则(?)为半环,称之为V-半环的强右正规幂等半环.利用这一结构证明了正规的型A-幂等半环是左零幂等半环的强右正规幂等半环及相关推论.与它平行地构造了V-半环的伪强右正规幂等半环,由这一结构证明了加法正规的型B-幂等半环为矩形半环的伪强半格幂等半环,也为左零半环的伪强右正规幂等半环,及相关推论.主要结论如下:
定理3.2.4半环S为正规的型A-幂等半环当且仅当S为左零幂等半环的强右正规幂等半环.
推论3.2.5半环S为[左正规,矩形,左零]型A-幂等半环当且仅当S为左零幂等半环的强[半格,右零,平凡]幂等半环.
定理3.2.8 S为正规型A-幂等半环和含幺带环的直积当且仅当S为型A-左环的强右正规幂等半环.
定理3.3.6设S为型B-幂等半环,则S为加法正规幂等半环当且仅当S是矩形半环的伪强半格幂等半环.
定理3.3.9 S为加法正规的型B-幂等半环当且仅当S为左零半环的伪强右正规幂等半环.
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∈D)S_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 (C_3), (C_4).
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 S~0 = {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→L_1,Π_(α∈D)ρ_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 L_1.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|_(S_α), 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,τ) is a congruence pair, then the relation:is a semiring congruence on S. Moreover , Kerρ_((N,τ)) = N,trρ_((N,τ)) =τ,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,τ_α)}_(α∈D), a family of (?)-normal congruencepairs of S, (N,τ) is defined by :
τ= {(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,τ) is a congruence pair of S, andρ_((N,τ)) = {(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,τ) |_(S_α)=ρ_((N_α,τ_α)).
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 (R_1),(R_2), 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.
引文
[1] Bandelt.H.-J.and M.Petrich. Subdirects of Rings and Distributive Lattice [J]. Proc. Edinburgh. Math. Soc, 1982, 25: 155-157.
[2] Shamik Ghost. A Characterization of semirings Which Are Subdirect Products Of Distributive Lattice and a Ring [J]. Semigroup Forum, 1999, 59(1): 106-120.
[3] Petrich,M. Lectures in semigroups [J]. Pitman: New York, 1976.
[4] Ren, X. M and Shum ,K.P. On generalized orthdogroups [J]. Comm. in Algebra, 2001, 29(6): 2341-2361.
[5] H. Mitsch. Subdirect Products of E-inversive Semigroups [J]. Austral. Math. Soc, 1990, 48: 66-78.
[6] Li Gang, Ding Yin. Ehresmann-typed wrpp semigroups [J]. Journal of shandong Normal University[J], 2005, 9, 20(3): 4-6,.
[7] Harry Hutchins. Division Semirings With 1+1=1 [J]. Semigroup Forum 1981, 22: 181-188.
[8] He Yong. Some studies on regular semigroups and generalized regular semigroups [J]. Ph.D.Theses, Zhong shan University, China, 2002.
[9] EI-Quallali. A., Structure theory for abundant and related semigroups [J].Ph.D.Thesis,York, 1980.
[10] P.J.Allen. A Fundamental Theorem of Homomorphisms for Semirings,Proc [J]. Amer. Math. Soc, 1969, 21: 412-416.
[11] John Zeleznekow. Regular Semirings [J]. Semigroup Forum, 1981, 23: 119-136.
[12] Petrich, M. Congruences on Inverse Semigroups [J]. Algebra, 1978, 55: 231-256.
[13] Warn R.J. On the Structure of semigroups wuich are union of groups [J]. Tran. Amer. Math. Soc, 1973, 186: 113-128.
[14] H. Melvin. Ideals in Semirings with Commutative Addition [J]. Notices Amer. Math. Soc, 1968, 5:321.
[15] PASTIJNF. Idempotent distributive semirings Ⅱ [J]. Semigroup Forum, 1983, 26(2): 151-166.
[16] S.H.Li and S.Z.Li. Ring Congruences on Common Commutative Regular Semiring [J]. Math. Sci Res. J., 2002, 6(5): 254-262.
[17] X.Zhang. Structure of Idempotent semiring Which Satisfies the Identity a + ab = a + b [J]. Math.Theory and Appli., 2002, 22(3): 29-33.
[18] X.Zhang and S.Z.Li. Structure of Idempotent Semiring Which Satisfies the Identity ab + b = a + b [J]. Math. Sci.Res. J., 2002, 6(9): 428-440.
[19] John M.Howie. Fundamentals of Semigroup Theory [M]. Oxford University Press Inc, Oxford, 1995.
[20] 左孝陵,李为鉴,刘永才. 离散数学[M].上海科学技术出版社, 1982.
[21] Petrich,M. Inverse Semigroup [M]. Wiley: New York, 1984.
[22] Petrich M, Relliy N R. Completely Regular Semigroups [M],New York: A Wiley-Intersience Publication, 1999: 1-50.
[23] 张璇. 幂等半环及其相关结构[D].山东:山东师范大学, 2003.
[24] 陈兆英. 关于逆半群的半格[D].山东:山东师范大学, 2004.
[25] 申冉. 关于半群的强半格上的同余[D],山东:山东师范大学, 2003.
[2] Shamik Ghost. A Characterization of semirings Which Are Subdirect Products Of Distributive Lattice and a Ring [J]. Semigroup Forum, 1999, 59(1): 106-120.
[3] Petrich,M. Lectures in semigroups [J]. Pitman: New York, 1976.
[4] Ren, X. M and Shum ,K.P. On generalized orthdogroups [J]. Comm. in Algebra, 2001, 29(6): 2341-2361.
[5] H. Mitsch. Subdirect Products of E-inversive Semigroups [J]. Austral. Math. Soc, 1990, 48: 66-78.
[6] Li Gang, Ding Yin. Ehresmann-typed wrpp semigroups [J]. Journal of shandong Normal University[J], 2005, 9, 20(3): 4-6,.
[7] Harry Hutchins. Division Semirings With 1+1=1 [J]. Semigroup Forum 1981, 22: 181-188.
[8] He Yong. Some studies on regular semigroups and generalized regular semigroups [J]. Ph.D.Theses, Zhong shan University, China, 2002.
[9] EI-Quallali. A., Structure theory for abundant and related semigroups [J].Ph.D.Thesis,York, 1980.
[10] P.J.Allen. A Fundamental Theorem of Homomorphisms for Semirings,Proc [J]. Amer. Math. Soc, 1969, 21: 412-416.
[11] John Zeleznekow. Regular Semirings [J]. Semigroup Forum, 1981, 23: 119-136.
[12] Petrich, M. Congruences on Inverse Semigroups [J]. Algebra, 1978, 55: 231-256.
[13] Warn R.J. On the Structure of semigroups wuich are union of groups [J]. Tran. Amer. Math. Soc, 1973, 186: 113-128.
[14] H. Melvin. Ideals in Semirings with Commutative Addition [J]. Notices Amer. Math. Soc, 1968, 5:321.
[15] PASTIJNF. Idempotent distributive semirings Ⅱ [J]. Semigroup Forum, 1983, 26(2): 151-166.
[16] S.H.Li and S.Z.Li. Ring Congruences on Common Commutative Regular Semiring [J]. Math. Sci Res. J., 2002, 6(5): 254-262.
[17] X.Zhang. Structure of Idempotent semiring Which Satisfies the Identity a + ab = a + b [J]. Math.Theory and Appli., 2002, 22(3): 29-33.
[18] X.Zhang and S.Z.Li. Structure of Idempotent Semiring Which Satisfies the Identity ab + b = a + b [J]. Math. Sci.Res. J., 2002, 6(9): 428-440.
[19] John M.Howie. Fundamentals of Semigroup Theory [M]. Oxford University Press Inc, Oxford, 1995.
[20] 左孝陵,李为鉴,刘永才. 离散数学[M].上海科学技术出版社, 1982.
[21] Petrich,M. Inverse Semigroup [M]. Wiley: New York, 1984.
[22] Petrich M, Relliy N R. Completely Regular Semigroups [M],New York: A Wiley-Intersience Publication, 1999: 1-50.
[23] 张璇. 幂等半环及其相关结构[D].山东:山东师范大学, 2003.
[24] 陈兆英. 关于逆半群的半格[D].山东:山东师范大学, 2004.
[25] 申冉. 关于半群的强半格上的同余[D],山东:山东师范大学, 2003.