Eventually regular Dubreil-Jacotin semigroups. (Q1764621)

An ordered semigroup, as defined here, is one with an order with respect to which every translation is isotone. An ordered semigroup \(S\) is a Dubreil-Jacotin semigroup if and only if there is an ordered group \(G\) and an isotone epimorphism \(f\colon S\to G\) that is principal, in the sense that the pre-image of the negative cone of \(G\) is a principal down-set of \(S\). Finally, a semigroup is eventually regular if each element has a regular power. The purpose of the paper is to derive necessary and sufficient conditions for an ordered eventually regular semigroup \(S\) to be Dubreil-Jacotin; when this is the case \(S\) has a greatest idempotent.