Study On Regularity Of Semirings And Their Related Problems

In this dissertation, the theory and methods which are applied to research semigroups to investigate the regularities of semirings; the semidirect product of semirings is introduced and its feature is described; finally the properties of without interactive decomposition of quasi-regular semigroups are discussed.In the first part,the characteristics of completely regular semirings are researched, the equivalent characterizations of distributive semirings whose sets of multiplication idempotents are left zero bands,rectangular bands or normal bands are obtained.In the second part, the definition of semidirect product of semirings is defined by structuring mapping.Besides, the equivalent characterizations of this kind of semirings when they are regular,orthodox, inverse monoids or quasi-regular semirings are discussed.In the third part, the definition of without interactive decomposition of quasi-regular semigroups is defined.For the left(right) quasi-inverse semigroups, the properties of *Green relations* *()L R about quasi-regular union decomposition are discussed. In particular, the properties of quasi-inverse and quasi-regular semigroups on this decomposition are obtained.