Structures On Idempotent Semirings

In this paper, we discuss the structures of idempotent semirings mainly. In the first chapter, we define a structure of the strong right normal idempotent semiring of V- semirings. That is, when A is a right normal idempotent semiring, {Sa : a A} be a collection of pairwise disjoint V-semirings, where V is a class of semirings, suppose that for each and , there exists a semiring homomorphism , satisfying conditions (C1) and (C2), and define two binary operations on the set by for any o, b S, suppose that , letThen (S, , o) is a semiring. We call it a strong right normal idempotent semiring of V-semirings. And by this we have the structure of the normal idempotent semiring which satisfies the identity ab + b = a + b arises as a strong right normal idempotent semiring of left zero idempotent semirings, and some corollaries. In the second chapter, we give the definition of the pseudo-strong right normal idem-potent semiring of V?semirings. And we have the additive normal idempotent semiring which satisfies the identity a + ab = a + b arises as a pseudo-strong right normal idempotent semiring of left zero semirings. And in the last chapter, we also have the idempotent semiring which satisfies the identity a + ab+a = a + b is an additive normal idempotent semiring, if and only if it is a pseudo-strong right normal idempotent semiring of left zero semirings, and other corollaries. Main conclusions:Lemma 1.2.1 Let A be a right normal idempotent semiring, take a collection of pairwise disjoint V-semirings {Sa : a A}, satisfying conditions (C1) and (C2) as above, and define two binary operations on the set S = Sa byfor any a, b S, suppose that a Sa, b SB, a, B A, letThen (S, , o) is a semiring.Theorem 1.2.5 A semiring S is a normal A- idempotent semiring, if and only if S is a strong right normal idempotent semiring of left zero idempotent semirings.Theorem 1.2.9 S is a direct product of a normal A- idempotent semiring and a commutative ring with an identity 1, if and only if S is a strong right normal idempotent semiring of A- left rings.Theorem 1.3.3 5 is an A-idempotent semiring, then 5 is a normal idem-potent semiring, if and only if S is a strong semilattice idempotent semiring of rectangular idempotent semirings.Lemma 2.2.1 Let A be an additive right normal idempotent semiring, take a collection of pairwise disjoint V- semirings {Sa : a A}, satisfying conditions (D1) and (D2) as above, and define two binary operations on the set S = Saby for any a, b S, suppose that a Sa, b SB, a, B A, let Then (S, ) is a semiring.Theorem 2.2.4 A semiring S is an additive normal C- idempotent semiring, if and only if S is a pseudo-strong right normal idempotent semiring of left zero semirings.Theorem 2.2.7 S is a direct product of an additive normal C- idempotent semiring and a ring, if and only if 5" is a pseudo-strong right normal idempotent semiring of left rings.Theorem 3.3 S is a "D- idempotent semiring, then S is an additive normal idempotent semiring, if and only if S is a pseudo-strong right normal idempotent semiring of left zero semirings.Theorem 3.6 S is a direct product of an additive left normal D- idempotent semiring and a ring, if and only if 5 is a pseudo-strong lattice idempotent semiringof left rings.