On left quasi-ample semigroups with ΠL¹-embeddable band of idempotents

ABSTRACT

Let ΠL¹ denote a direct power of L¹, the two-element left zero semigroup with identity adjoined. A semigroup S is called left quasi-ample if for each a∈S there exists a unique idempotent a⁺∈S such that $xa=ya\iff x{a}^{+}=y{a}^{+}$ for all x, y∈S¹ and the left ample condition $e^2=e\Rightarrow {\left(ae\right)}^{+}a=ae$ holds. Generalizing a recent result in [3], we prove that the semigroups in the title are embeddable into certain transformation semigroups. Our embedding provides an easy way to construct (finite) proper covers for (finite) such semigroups. Moreover, we show that each proper such semigroup is embeddable into a semidirect product of a ΠL¹-embeddable band by a right cancellative monoid, giving a partial answer to a question raised in [1].

KEYWORDS:

• Left regular band
• left quasi-ample semigroup
• proper cover

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 20M19

Acknowledgement

This research was supported by Chiang Mai University.