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

Abstract

© 2017 Taylor & Francis. Let ΠL1denote 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 ⇔ xa+= ya+for all x, y∈S1and the left ample condition e2= e ⇒ (ae)+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].