Journal of the
Korean Mathematical Society JKMS

A semigroup $S$ is called a weakly abundant semigroup if its every $\widetilde{\mathcal{L}}$-class and every $\widetilde{\mathcal{R}}$-class contains an idempotent. Our purpose is to study an analogue of orthodox semigroups in the class of weakly abundant semigroups. Such an analogue is called a left quasi-abundant semigroup, which is a weakly abundant semigroup with a left quasi-normal band of idempotents and having the congruence condition (C). To build our main structure theorem for left quasi-abundant semigroups, we first give a sufficient and necessary condition of the idempotent set $E(S)$ of a weakly abundant semigroup $S$ being a left quasi-normal band. And then we construct a left quasi-abundant semigroup in terms of weak spined products. Such a result is a generalisation of that of Guo and Shum for left semi-perfect abundant semigroups. In addition, we consider a type $Q$ semigroup which is a left quasi-abundant semigroup having the PC condition.

Keywords: weakly abundant semigroups, weak spined products, left quasi-abundant semigroups, type $Q$ semigroups

MSC numbers: 20M10

Supported by: The research is supported by Natural Science Foundation of China (Grant No: 11471255, 11501331). The third author was supported by Shandong Province Natural Science Foundation (Grant No: BS2015SF002), SDUST Research Fund (No. 2014TDJH102), and Joint Innovative Center for Safe and E?ective Mining Technology and Equipment of Coal Resources, Shandong Province.