【作者】 王锐；

【导师】 李刚；

【作者基本信息】 山东师范大学 ， 基础数学， 2011， 硕士

【摘要】 本文主要分为两大章节，第一章节讨论的是加法含零双半环的分配格的结构；第二章节给出了三类幂等双半环，并刻划了它们的结构及其一系列相关问题，具体内容如下：第一章：第一节，给出了引言和预备知识．第二节，给出加法含零双半环的分配格的定义并刻划了它的结构．设D为分配格，∈D}为一族两两非交的加法含零双半环，令且若满足：则称S为加法含零双半环的分配格，记为S={D；sa)，且(S，+，;*)为双半环．主要结论如下：定理1．2．3设S={D；&)，若满足则P为S上的双半环同余，且S为分配格D和双半环s／卢的拟次直积；反之，若S={D，&)上存在形如(1)定义的同余P，且则S满足(0)，(Cs)定理1．2．4设S={D；sa)若则定理1．2．5设(S+，，+)为加法含零双半环，记零元为0，且0为(s，)的乘法中心元，即Vn∈S 0n=n0在s定义关系则n为s上的双半环同余，且(s/n,+)为半格．第二章首先构造了v双半环的强右正规带的结构，即令^为右正规幂等双半环，{sa/a∈^)为一族两两非交的V双半环，V表示双半环类，，存在sa到sb的双半环同态氐，p，即满足(R1)，(R2)，在集合定义二元运算设则为双半环，称为V双半环的强右正规带．利用这一结构证明了乘法””正规的型A幂等双半环是左零幂等双半环的强右正规带及其相关推论．与它下行地构造了v双半环的伪强右正规带，由这一结构证明了加法正规的型B幂等双半环为矩形双半环的伪强半格及其相关推论，最后类似地构造了v双半环的强半格，利用这一结构证明了乘法正规型G可分配双半环是矩形双半环的强半格及其相关推论，主要结论如下：定理2．2．5设双半环s为乘法正规的型A幂等双半环当且仅当s为左零幂等双半环的强右正规带．推论2．2．6双半环s为[左正规，矩形，左零]型A幂等双半环当且仅当s为左零幂等双半环的强[半格，右零，下凡]带．定理2．2．8若s为正规的型A幂等双半环和含幺幂等双半环的拟直积则s为型A左幂等双半环的强右正规带．定理2．2．9若s为左正规矩形，左零]的型A幂等双半环和含幺幂等双半环的拟直积则s为型A左幂等双半环的强半格[右零，平凡]带．定理2．3．5双半环s为型B幂等双半环，s为加法正规的幂等双半环当且仅当s为矩形双半环的伪强半格．定理2．4．5设s为型G可分配双半环，则s是乘法正规的，当且仅当s是矩形双半环的强半格．推论2．4．6若s为型G可分配双半环，且1∈s，则s是乘法左(右)正规的，当且仅当s是矩形双半环的强半格．定理2．4．8若s是乘法正规的型C-可分配双半环和乘法半群为带的含幺双半环的拟直积，则s是R-双半环的强半格．推论2．4．9若s是乘法左(右)正规的型c-可分配双半环和乘法半群为带的含幺双半环的拟直积，则s是R-双半环的强半格.更多还原

【Abstract】 The dissertation is divided into two chapters.In chapter l,we mainly discuss the structures of a distributive lattice of bi-semirings with additional zero elements;in chapter 2,we give three kinds of idempotent bi-semirings and other structures.The results are given in follow.In the first part of Chapter l,we give the introduction and preliminaries.In the second chapter of chapter 1, we mainly give the definition of a distributive lattice of bi-semirings with additional zero elements,and give a characterization of a structures on it.Let D be a distributive lattice,(?)are a collection of pairwise disjoint bi-semirings with additional zero elements.LetS = (?,,and(?),If S satisfies conditions:(?)Then we call S a distributive lattice of bi-semirings with additional zero elements(?), written it as(?),and(/)is a bi-semiring.Main results:Theorem 1.2.3 Let S = {D;Sa},if. S satisfies the conditioner every (?) a relation^ on S is defined by Then p is a bi-semiring congruence on S,and S is a subdirect product of a distributive lattice D and a bi-semiring Sl/p;Conversely,if there exists the same congruence as(l)on S,and (?)Then S satisfies (C4), (C5).Theorem 2.2.4 Lets’ =(?).Theorem 2.2.5 Let(?))is a bi-semiring with additional zero elements.Zero elements are written as 0,and 0 is the multiplicative central element of (s,?).That is (?) a relation n on S is defined by(?)Then r/ is a bi-semiring congruence on S,and (S/r/, +) is a semilattice.In chapter 2,firstly we define a structure of the strong right normal band of V—bi-semirings.That is,whenA is a right normal idempotent bi-semiring,{Sa \ a￡A} are a collection of pairwise disjoint V—bi-semirings, where V is a class of bi-semirings,suppose that for each (?),there exists a bi-semiring homomorphism(?)satisfying conditions(R1i), (R2),&nd define two binary operations on S = (?), suppose that (?) is a bi-semiring.we call it a strong right normal band of V—bi-semirings. Andby this we have the structures of the normal Type A—idempotent bi-semiring which arises as a strong right normal band of left zero idempotent bi-semirings,and some corollaries. Secondly,we give the definition of the pseudo-strong right normal band of V—bi-semirings.And we have the additive normal Type B—idempotent bi-semiring which arises as a pseudo-strong semilattice of rectangular bi-semirings,and some corollaries.Thirdly,we define a structure of the strong semilattice of V—bi-semirings.And by this we have the structures of the distributive TypeC—bi-semiring which arises as a strong semilattice of rectangular bi-semirings,and some corollaries.Main results:Theorem 2.2.5 A bi-semiring S is a normal Type A—idempotent bi-semiring,if and only if S is a strong right normal band of left zero idempotent bi-semirings. Corollary 2.2.6 A bi-semiring S is a [left normal,rectangular,left zero]Type A—idempotent bi-semiring,if and only if S is a strong [semilattice,right zero,trivial] band of left zero idem-potent bi-semirings.Theorem 2.2.8 If S is a pseudo-direct product of a normal Type A—idempotent bi-semiring and a idempotent bi-semiring with an identity l,then S is a strong right normal band of Type A—left idempotent bi-semirings.Corollary 2.2.9 If S is a direct pseudo-product of a left normal[rectangular,left zero] Type A—idempotent bi-semiring and a idempotent bi-semiring with an identity l,then S is a strong semilatticefright zero,trivial]band of Type A—left idempotent bi-semirings.Theorem 2.3.5 S is an Type B—idempotent bi-semiring,then S is an additive normal idempotent bi-semiring, if and only if S is a pseudo-strong semilattice of rectangular bi-semirings.Theorem 2.4.5 S is a distributive Type C—bi-semiring,then S is a multiplicative normal bi-semiring if and only if S is a strong semilattice of rectangular bi-semirings.Corollary 2.4.6 S is a distributive Type C—bi-semiring with an identity l,then S is a multiplicative left(right) normal bi-semiring if and only if S is a strong semilattice of rectangular bi-semirings.Theorem 2.4.8 If S is a pseudo-direct product of multiplicative normal and distributive Type C—bi-semiring and a bi-semiring, which has a unit element and a multiplicative semigroup band,then S is a strong semilattice of R-bi-semiring.Corollary 2.4.9 If S is a pseudo-direct product of multiplicative left(right)normal and distributive Type C—bi-semiring and a bi-semiring, which has a unit element and a multiplicative semigroup band,then S is a strong semilattice of R-bi-semiring.更多还原