Idempotent pure extensions by inverse semigroups via quivers
A quiver consists of a set of objects and, for any objects \(u\) and \(v\), of a set of arrows from \(u\) to \(v\). On the set of arrows a partial binary operation \(+\) is defined and this yields a structure which is like a category except that identities need not exist, and addition of non-consecutive arrows is possible. Given an inverse monoid \(T\) and a partial action of \(T\) on a quiver \(\mathcal C\) which has object set \(T\), the authors define the notion of an inverse \(T\)-quiver. To each inverse \(T\)-quiver \(\mathcal C\) the authors associate in a canonical way an inverse semigroup \(S({\mathcal C})\). If in particular for every object \(a\) of \(\mathcal C\) the set of arrows from \(a\) to \(a\) constitutes a semilattice for the operation \(+\), then there exists a naturally defined idempotent pure congruence \(\delta\) on \(S({\mathcal C})\) such that \(S({\mathcal C})/\delta\) is embeddable into \(T\), i.e., \(S({\mathcal C})\) is an idempotent pure extension by an inverse subsemigroup of \(T\). Conversely, if \(S\) is an inverse semigroup which is an idempotent pure extension by an inverse senigroup \(T\), then a so-called derived quiver of the extension is constructed. It is proven that this derived quiver \(\mathcal D\) is a \(T^1\)-quiver with \(S=S({\mathcal D})\). The paper contains several variants of these results. In the last two sections, results by O'Carroll and Billhardt concerning idempotent pure extensions by inverse semigroups are derived.
scientific article; zbMATH DE number 1308155
scientific article