Gabriel topologies on coherent quantales (Q1295504)

The notion of a Gabriel topology on a quantale \(Q\) was first investigated by the reviewer in his book ``Quantales and their applications'' [\textit{K. I. Rosenthal}, Pitman Res. Notes Math. 234 (1990; Zbl 0703.06007)]. This paper considers the frame of Gabriel topologies on a coherent quantale \(Q\) (recall that if ``\&'' denotes the operation on \(Q\), then \(Q\) is coherent iff its top element is finite and its finite elements are closed under ``\&''). These ideas are applied to ring theoretic spectrum constructions, and the compact Gabriel topologies are also related to a system of propositional logic. In this context, local compactness of the frame of compact topologies corresponds to a completeness theorem for this logical system. This paper will be of interest to categorical readers interested in algebra, because of its connections with ring theory, as well as logic (the logic involved in this setting is a so-called ``substructural'' logic).