Reflectors to quantales

TL;DR

This paper demonstrates that marked quantales can be reflected into quantales by constructing free quantales over them, broadening the understanding of their structure and relationships.

Contribution

It introduces a reflection process from marked quantales to quantales via free constructions, unifying various classes like posemigroups and prequantales.

Findings

Constructed free quantales over marked quantales.

Established a reflection from marked quantales to quantales.

Discussed reflection via injective hulls of posemigroups.

Abstract

In this paper, we show that marked quantales have a reflection into quantales. To obtain the reflection we construct free quantales over marked quantales using appropriate lower sets. A marked quantale is a posemigroup in which certain admissible subsets are required to have joins, and multiplication distributes over these. Sometimes are the admissible subsets in question specified by means of a so-called selection function. A distinguishing feature of the study of marked quantales is that a small collection of axioms of an elementary nature allows one to do much that is traditional at the level of quantales. The axioms are sufficiently general to include as examples of marked quantales the classes of posemigroups, $\sigma$-quantales, prequantales and quantales. Furthermore, we discuss another reflection to quantales obtained by the injective hull of a posemigroup.