Completions of posemigroups by cuts and beyond

Abstract

This work is devoted to describing the completion of a posemigroup by cuts. We introduce cut-stable morphisms between posemigroups and obtain that the category \({\mathsf {RQuant_{\wedge }}}\) of quantales with meet and residuals preserving morphisms is a full reflective subcategory of the category \({\textsf{CSPoSgr}}\) of posemigroups with cut-stable morphisms. As an application, we also characterize other kinds of completions for posemigroups by means of lower cuts.