Structures And Some Congruences On Bisemirings

In this dissertation, we give a characterization of the bisemiring congruences and discuss three Green relations on the idempotent bisemiring. Then we study some structures of bisemirings through some bisemiring congruences. The main results are given as follows:In Chapter 1, we give the introduction and preliminaries.In Chapter 2, we mainly give a characterization of bisemiring congruences and get the least bi-ring congruence and the least additive inverse bisemiring congruence on some bisemirings in the first section. In other three sections we study the Green D-relation, Green R-relation and Green L-relation, and get the necessary and sufficient conditions of D+∩D*∩D?, R+∩R*∩R?and L+∩L*∩L? relations satisfied by elements of the idempotent bisemirings. We characterize them as the congruences on the idempotent bisemirings. Finally we get the Mal' cev product decomposition of the corresponding subvarieties.In Chapter 3, we mainly study the pseudo-strong distributive lattice of bisemiring varieties and get the pseudo-subdirect product decomposition in the first section. In the second section we discuss the structure of the addition left normal bisemirings, which satisfies the identities a + ab+a=aand a + ba+a=a. And we have the direct product decomposition of some bisemiring.In Chapter 4, the pseudo-strong semilattice of the bisemiring varieties is studied. We discuss the structure of the multiplication normal bisemiring, which satisfies the identities aba + bab=a+band aba ? bab=a?bin the first section. And in the second section we discuss the structure of the multiplication left normal bisemirings, satisfying the identities ab + ba=a+band ab ? ba=a?b. Based on them we further get the direct product decomposition of two kinds of bisemirings.