On semidirect products of quantale enriched monoids

TL;DR

This paper explores the structure of semidirect products in quantale enriched monoids, introducing split extensions that generalize classical concepts and analyzing their properties and applications in preordered monoids.

Contribution

It introduces a new class of split extensions for quantale enriched monoids and connects them to semidirect products, extending classical monoid theory into a quantale-enriched setting.

Findings

Generalized Schreier split extensions for quantale enriched monoids

Characterization of semidirect products in this enriched context

Application to preordered monoids and comparison with existing theories

Abstract

We consider monoids equipped with a compatible quantale valued relation, to which we call quantale enriched monoids, and study semidirect products of such structures. It is well-known that semidirect products of monoids are closely related to Schreier split extensions which, in the setting of monoids, play the role of split extensions of groups. We will thus introduce certain split extensions of quantale enriched monoids, which generalize the classical Schreier split extensions of monoids, and investigate their connections with semidirect products. We then restrict our study to a class of quantale enriched monoids whose behavior mimics the fact that the preorder on a preordered group is completely determined by its cone of positive elements. Finally, we instantiate our results for preordered monoids and compare them with existing literature.