On Superabundant Semigroups And Its Subclasses

The research of regular semigroups and its subclasses are dominant in the study of semigroup theory. As the development of semigroup theory, many domestic and foreign famous scholars have started working some classes of generalized regular semigroups. And a series of achievements are obtained. Especially, the research of superabundant semigroups and its subclasses has formed an impotant subject of study of generalized regular semigroups.In this paper, we concentrate on superabundant semigroups and three special classes of superabundant semigroups, that is, cryptoabundant semigroups, cryptoabundant semigroups in which its idempotents are central and cryptoabundant semigroups in which the generalized Green’s relation ??and ?? are congruences.Firstly, we discuss superabundant semigroups by using axiomatic conditions. an equivalent definition of superabundant semigroups is given. A semigroup S is superabundant semigroup if and only if S satisfies(C1)-(C4) of axiomatic conditions. The result brings convenience for the study of superabundant semigroups.Secondly, we focus on cryptoabundant semigroups. And the definition of cryptoabundant semigroups is given. Superabundant semigroup S is called a cryptoabundant semigroup if the generalized Green’s relation ?? is congruence on S. On this basis, we focus on the discussion of some properties of cryptoabundant semigroups, and give several necessary and sufficient conditions for judging cryptoabundant semigroups. cryptoabundant semigroups is a special kind of superabundant semigroups.Finally, we study two special classes of cryptoabundant semigroups, that is, cryptoabundant semigroups in which its idempotents are central and cryptoabundant semigroups in which the generalized Green’s relation ??and ?? are congruences. And several necessary and sufficient conditions for judging the two special classes of cryptoabundant semigroups are given, respectively.