Structures And Congruences On Some Semirings

In this paper we mainly discuss structures and congruences on some semirings. First,according to the definition of strong distributive lattice of semirings, we define the pseudo-strong distributive lattice semiring S and the pseudo-direct product,and we prove that the pseudo-direct product that we just define is a semiring, then we prove that S is the pseudo-sub direct product of D and S/9. In the second section ,we give all the ring congruences on a commutative regular semiring S and show that from the lattice of all full,closed,ideal subsemirings of S to the lattice of all the ring conruences on 5,there is a lattice isomorphism. Moreover we generize a result of Howie [12] to a commutative regular semiring. And we describe the join r V p of a ring congruence r and an arbitrary congruence p on S. Similarly,we discribe all the divisible semiring congruences on a distributive semiring. At last,we give the least distributive lattice congruence on a commutative distributive semiring and an idempotent distributive semiring. Main results are following:Theorem 1.9 Let 5 is a- pseudo-strong distributive lattice semiring ,0 is a congruence of the definition in Lemma 1.4. Then S is the the pseudo-sub direct product of D and S/B.Theorem 2.15 The map RC(S) a +b 6 is a lattice isomorphism.Theorem 2.18 For any congruence p and a ring congruence T on 5", a(r V (x + a)p(y + b) ,here x,y kerr Particularly if a is the least ring congruence on 5, then a(a V p)b,here e, / e E^(S).Theorem 2.21 For any congruence p and a ring congruence T on S,T V p is ring congruence, furmore ar V p}ba + bker(r VTheorem 3.17 The map DSC(S) I = {(a, 6) G 5 x 5 : ab1 G 7, 6' G V(b)}Theorem 3.18 For any divisive semiring congruence on p on 5, p = C. Then the following are equivelent:Theorem 4.4 Let is a commutative semiring,define a relation a on S as: a, 6 G S,acrb /. Then is the least distributive lattice congruence on S.Theorem 5.4 Let 5 is an idempotent distributive semiring,difine a relation as: a, 6 G S,aab . Then a is the least distributive lattice congruence on 5.