The Generalization Green Relations And Its Application

The semigroup is a kind of weakening of group, only requests the dual operation tosatisfy the associative law. The rise of the researchment of semigroup date from 1960s,however, semigroup theories are similar to the theory of group and ring in some parts.The most important achievement mainly attributes to Rees, Cli?ord and Dubreil at theinitial period. As to the 70's, the semigroups theory develops and enriches rapidly ,such as Cli?ord, Petrich M and Howie J B, whose work are of profound theoretical andpractical significance, the content involves with congruence, structure, varieties and so on.In the beginning, it's depend on Green relations to sutdy some special semigroups, justlike Cliford semigroups, inverse semigroups, oxthodox semigroups, regular semigroups,complete regular semigroups and so on.The usural Green relations are generalized to ?-Green relations and the definitionof abundant semigroups are given by Fountain J in 1979. And then, the researchmentof abundant semigroups is become more and more refinement and systematic, and itbecomes a very popular field of the algebraic theorey of semigroups,such as superabundantsemigroups, semisuperabundant semigroups and H#-abundant semigroups and so on.In this thesis, we mainly study regular semisuperabundant semigroups and the ap-plication of the decomposition of quasi strong semilattice,which can be divided into threechapters.In chapter 1, we introduce some basic knowledge of semigroups, the background anddevelopment of semigroup, especially of abundant semigroup, and gives some basic con-ceptions and preliminaries theory. What's more should be mentioned is that we introducetwo of the most important relations, that is equivalence and congruence. At last, we givetwo generalizations of strong semilattice and a brief introduction.In chapter 2, we study the structure of normal semisuperabundant semigroup. In thebeginning, we introduce the most important conception,ρ-Green relations, and talk oversome properties of it. And then,the structure theorm of semisuperabundant semigroupnormal semisuperabundant semigroup are given. Last but not the least, we prove that a semisuperabundant semigroup is a normal semisuperabundant semigroup if and only if itis a strong semilattice of completely Jρ-simple semigroups.In the last chapter, the problem of structure of regular bands and the structure ofnormal crptosemisuperabundant semigroup are studyed. By the method of quasi strongsemilattice of semigroups, we prove that a band is a regular band if and only if it is aquasi strong semilattice of rectangular bands, it generalizes the result of Yamadda andKimura on normal bands, that is a band is a normal band if and only if it is a strongsemilattice of rectangular bands. At last, the theorm of a semisuperabundant semigroupis a normal crptosemisuperabundant semigroup if and only if it is a quasi strong semilat-tice of completely Jρ-simple semigroup is proved, which generalizes the result of Clii?ordand Petrich on completely regular semigroups.