| Semigroup theory, although the study originated in the group,its establishment from research objects to research methods with the group has a great deal of difference, almost no common ground between the two.Conducted by internal mathematics (such as operator theory,topology,probability theory and etc) and external mathematics (especially computer science) , Semigroup theory has been researching systemly for almost sixty years. In recent decades, particularly in emerging disciplines, the needs of the development such as formal language and automata theory, Code theory, makes the development of semigroup theory very quickly.One of the researching importance of semigroup theory is the structure of semigroup. Presently the structure research of semigroup which is abundant is regular semigroup. And the key means of structure researching of regular semigroup is Green relation which puts forward by J.A.Green in 1951.Popularizing Green relation has a practical significance.But to non-re- gular semigroups, the role of Green relation is not significant.The text based on equivalence relation and congrue frequently-used,defined generalized Green relation,and studied related property of Green relation. In this thesis, the structure of regular H (?)-cyber group and the properties and the structure of LÏC- regular semigroup are researched by using the structure of knitted semilattice andÏ(?)Green relations.1. In chapter one, first we briefly introduce the background and the development of semigroups theory, the current research at home and abroad as well as the acquisition of relevant achievements, and clarify some problems and the work to be done in this thesis.Then we introuduce some basic knowledge and related conception,two of the most important in the research is mentioned ,that is equivalence and congruence. In actual fact, the strong semilattice decomposition of semigroups is one of the best structural decompositions, at last we will give the definition and generalization of strong semilattice.2. In the second chapter we continue to generalize Green relations to a new Green relati- ons, that is Green (?)-relations from the original Green relations, *- Green relations, Green (?)- relations accompanied by new definitions of L(?) , R (?), H (?), D (?), J (?). We consider regular H (?)-cyber groups in the class of H (?)-abundant semigroups. By using knitted semilattice of semigroups, we give some structure theorems for regular H (?)-cyber groups, right quasi-nor- mal H (?)-cybergroups and normal H (?)-cyber groups. Our main result generalizes a classical theorem of Petrich- Reilly on normal cryptic groups from the class of regular semigroups to the class of generalized abundant semigroups and also entriches a recent result of Guo-Shum on left cyber groups.3. In the third chapter, we generalize Green relations to a new Green relations, that is Green (?)-relations by new definitions of L(?) , R (?), H (?), D (?), J (?). By utilizing generalizedÏ(?)Green relations on semigroups, we defined a generalized regular semigroups of LÏC- regular semigroups, and obtained some properties and structure. Some equivalent conditions for LÏC-regular semigroups are proved, a semigroup is an LÏC-regular semigroup if and only if it is a strong semilattice of L -cancellative monoids.4. In this chapter we discuss the set of all right cosets of a group. We show that the set is an inverse semigroup under a given multiplitation and research the properties of its idempotents. Moreover a necessary and sufference condition for it to be a Clifford semigroup are given. |
