Left Quasi-weakly Abundant Semigroups

A semigroup S is said to be weakly abundant if every (?)-class and every (?)-class of S contain an idempotent of S. The weakly abundant semigroups is a common generalization of the regular semigroups and the abundant semigroups. In recent years, the study of weakly abundant semigroups and their subclasses become one of the important areas in the study of semigroup theory. As is known to all, the properties of the set of idempotents play an important role in the algebraic structure of semigroups. Therefore, the study of weakly abundant semigroups of the set of idempotents to meet certain conditions raises widespread concern.In this paper, we mainly study a class of weakly abundant semigroups, the so-called left quasi-weakly abundant semigroups. A weakly abundant semigroup S is said to be left quasi-weakly abundant if S satisfies the congruence condition and its set of idempotents forms a left quasi-normal band(the band that satisfies the identity xyz = xyxz ). This kind of semigroup is the common generalization of the left quasi-regular semigroups in the class of regular semigroups and the left quasi-abundant semigroups in the class of abundant semigroups.This paper first defines the left quasi-weakly abundant semigroups, and obtains some basic properties of such semigroups. Based on this, by using the right normal band and the weakly abundant semigroups with simple structure, we establish an algebraic structure of left quasi-weakly abundant semigroups. It is proved that the left quasi-weakly abundant semigroups are isomorphic to the weak spined product of (?)-unipotent semigroup and the right normal band. In addition, also gives an equivalent description of left quasi-weakly abundant semigroups satisfying PC condition. It is proved that the left quasi-weakly abundant semigroup satisfying the PC condition is a left quasi-weakly abundant semigroups of type P if and only if the relation η is an admissible congruence on S and satisfies S/η is the (?)-unipotent semigroup. And the relation ξ is an admissible congruence on S and satisfies S/ξ is the (?)-unipotent semigroup, where η and ξ are defined as follows:η=﹛(x,y)∈S×S:((?)f∈E(y~+))x=yf﹜ξ=﹛(x,y)∈S×S:((?)e∈E(y~+))x=ey﹜...