An application of a theorem of Ash to finite covers.
The relation \(\widetilde R\) on a semigroup \(S\) is defined by \(a\widetilde Rb\) iff \(ea=a\Leftrightarrow eb=b\) for all \(e\in E(S)\). A semigroup \(S\) is called weakly left quasi-ample (wlqa), if for all \(a\in S\) the \(\widetilde R\)-class of \(a\) contains a unique idempotent \(a^+\) and the ample identity \(ae=(ae)^+a\) holds for all \(a\in S\), \(e\in E(S)\); \(S\) is weakly left ample (wla), if the idempotents of \(S\) commute and \(\widetilde R\) is a left congruence. The class of wlqa semigroups is a \((2,1)\)-quasivariety with binary semigroup operation and unary operation \(a\mapsto a^+\). This class contains the classes of groups, inverse, left ample and weakly left ample semigroups. It is shown that every finite wlqa semigroup has a finite proper wlqa cover and all wla semigroups have representations as \((2,1)\)-subalgebras of partial one-one transformations, where the unary operation maps a transformation \(\alpha\) to the identity map on the domain of \(\alpha\).
scientific article; zbMATH DE number 2153436
scientific article