Regular Superabundant Semigroups

A semigroup S is called an abundant semigroup,if every L*-class and every R*-class of S contain an idempotent.A semigroup S is called a superabundant semigroup,if every H*-class of S contains an idempotent.The abundant semigroups and the superabundant semigroups are natural extension of the regular semigroups and the completely regular semigroups,respectively.Recently,the research of abundant semigroups and their subclasses become a significant area in the study of semigroupsIn this paper,the research mainly focus on a special class of superabundant semigroups,which is called a regular superabundant semigroup.A superabundant semigroup S is called a regular superabundant semigroup,if the set of all idempotents of S forms a regular band.It's easy to see,a regular orthogroup in the class of a completely regular semigroup is a regular superabundant semigroup.So,a regular superabundant semigroup is a natural generalization of a regular orthogroup in the class of superabundant semigroupsFirstly,regular superabundant semigroups is defined in this paper.Also,we obtain some basic property of regular superabundant semigroups.Next,we establish the general structure of regular superabundant semigroups by introducing the concepts of twist product,and we prove that a semigroup S is a superabundant semigroup,if and only if S can be expressed as a twist product of a left regular band,a strongly semilattice of cancellative monoids and a right regular band.This result generalizes the structure theorem about the regular orthogroups which given by the famous semigroup expert M.Yamada.Moreover,we give a spined product structure of regular superabundant semigroups,and we prove that a semigroup S is a cyber-group,if and only if S can be expressed as a spined product of a left regular cyber-group and a right regular cyber-group.At the end of this paper,we provide an non-trivial example of regular superabundant semigroups based on the twist product of regular superabundant semigroups.