The Structure Of Some Bi-Semirings

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.