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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.
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참고문헌 (20)
참고문헌 신청
Mário J.J. Branco / 2018 / Ehresmann monoids: Adequacy and expansions / Journal of Algebra 513:344~367
A. El-Qallali / 1981 / Idempotent-connected abundant semigroups / Proceedings of the Royal Society of Edinburgh: Section A Mathematics 91(1-2):79~90
J. Fountain / 1982 / Abundant semigroups / Proc. London Math. Soc 44(1):103~129
John Fountain / 2009 / Membership of A ∨ G for classes of finite weakly abundant semigroups / Periodica Mathematica Hungarica 59(1):9~36
Xiaojiang Guo / 2005 / Left semi-perfect abundant semi-groups of type W / Proceedings of the Royal Society of Edinburgh: Section A Mathematics 135(3):603~614
A. El-Qallali / 1981 / Idempotent-connected abundant semigroups / Proceedings of the Royal Society of Edinburgh: Section A Mathematics 91(1-2):79~90
J. Fountain / 1982 / Abundant semigroups / Proc. London Math. Soc 44(1):103~129
John Fountain / 2009 / Membership of A ∨ G for classes of finite weakly abundant semigroups / Periodica Mathematica Hungarica 59(1):9~36
Xiaojiang Guo / 2005 / Left semi-perfect abundant semi-groups of type W / Proceedings of the Royal Society of Edinburgh: Section A Mathematics 135(3):603~614
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