Congruences On Two Classes Of Semirings

Congruences play important roles in the study of semiring algebra.In this dissertation,we investigate the relationship between congruences and convex normal subgroups on zerosumfree semirings whose multiplicative semigroups are0-groups,study the properties of flat congruences and weakly flat congruences on T-semialgebras,and obtain some properties of flat congruences.The main results are as follows:1.We characterize congruences of zerosumfree semirings whose multiplicative semigroups are 0-groups,and prove that there is a one-to-one correspondence between congruences and convex normal subgroups on zerosumfree semirings whose multiplicative semigroups are 0-groups.2.We study the congruences of the tropical semiring T,and obtain that there are only three congruences on T,which proves that T is subdirectly irreducible.3.We study the flatness and weak flatness of congruences on the tropical polynomial semiring T[X],discuss the relationship between flat congruences and weak flat congruences,and give the characterization of flat congruences generated by any binary relationship on T[X].