H~#-abundant Semigroups

"Semigroup Algebra Theory"which is implelled by computer science and information science and researched systemically for more than sixty years, has been become a distinctive and special subject embranchment of"algebra". It has many applications in the specific fields of formal language, automaton and so on. As is showed in the relation of"Semigroup theory"and"Group theory",it is genereral accepted that of"Ring theory"and"Field theory". This status are established not only the appearance of a passel of system research production, but also the form of a suit of particular system research thoughtway. Abundant semigroups rised in the 1970s, is a generalized regular semigroup with regular semigroup as its true sub- class. At present, the researchment of such semigroup has become more and more popular. In this thesis, we first introduce the definitions of semigroups proceed with properties, focusing on the properties and the semilattice decomposition of some H #?abundant semigroups. It can be divided into the following chapters:1.The first chapter of this thesis is the introduction. First, we briefly introduce the origin 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 in the second section, we listed some basic definitions and properties of se- migroups which maybe involved in this thesis, such as the definition of regular semigroup and completely regular semigroup, rectangular band, isomorphism, natural order and so on. As for congruence is an essential element for the study of the properties of semigroups, we introduce equivalence relations and congruence, and prove the theorem about congruence in section 3. In actual fact, an important purpose of the study of semigroups is to study its structure ,while the strong semilattice decomposition of semigroups is one of the best structural decompositio- ns. In order to facilitate the comparison, we will give the definition and generalization of strong semilattice in the last section.2. In the second chapter we continue to generalization Green relations to a new Green relations, that is # ?Green relations from the original Green relations, * ?Green relations, ~ ?Green relations accompanied by new definitions of L# , R #, H #, D #, J #, and introd- uce(left,right) # ?ideal and discuss a series of properties of # ?Green relations and # ?ideals. By using this new set of Green's #? relations, we study a class of semigroups be called H #? abundant semigroups(each H #?classes contains idempotants)and get a necessary and suffice- ient condition such that a semigroup is a H #?abundant semigroups. At last, we discuss a cla- ss of special H #?abundant semigroups, normal H #?abundant semigroups, and further study the properties of such semigroups.3. The chapter 3 is divided into three sections. The first one is prior knowledge, we intro- duce the definitions of superabundant semigroups, cryptogroups, completely regular semigro- ups and discuss the properties and structural decomposition theorem of completely regular se- migroups, and generalize some results of completely regular into H #?superabundant semigr- oups.In the second section, we study some strong semilattice at the foundation of strong semi- lattice we have known, and particularly study the structure of H #?superabundant semigroups A regular H #?superabundant semigroups whichGreen relations H is a congruence is regular H #?cryptosuperabundant semigroups. Finally, we study the properties and structures of reg- ular H #?cryptosuperabundant semigroups.4. As well known, we can construct a semigroup using the strong semilattice , strong se- milattice decomposition is one of the best structural decomposition in the structural decompo- sition of semigroups. At present, strong semilattice decomposition of semigroups has several generalization one of which is a refined semilattice of semigroups. The final chapter in which we study the refined semilattice structure of regular ortho H #?cryptoabundant semigroups c- ombined with the theorem of regular H #?cryptosuperabundant semigroups in chapter 3. At last, we prove that a completely regular semigroup S is a normal orthocryptogroup if and only if it is a strong semilattice of rectangular groups.