On left quasi-ample semigroups withΠL1-embeddable band of idempotents
Bernd Billhardt, Yanisa Chaiya, Ekkachai Laysirikul, Kritsada Sangkhanan, Jintana Sanwong · Communications in Algebra · 2017
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⇔xa+=ya+ for all x, y∈S1 and the left ample condition e2=e⇒(ae)+a=ae holds. Generalizing a recent result in [3 Billhardt, B., Laysirikul, E., Sangkhanan, K., Sanwong, J., Sommanee, W. (2016). On ℛ-unipotent semigroups with ΠL1-embeddable band of idempotents. Semigroup Forum 92:228–241.[Crossref], [Web of Science ®] , [Google Scholar]], 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 Auinger, K., Gomes, G. M. S., Gould, V., Steinberg, B. (2004). An application of a theorem of Ash to finite covers. Stud. Log. 78:45–57.[Crossref] , [Google Scholar]].