Gabriel topologies on coherent quantales
type: Item
instance of: scholarly article
title: Gabriel topologies on coherent quantales
zbMATH Open document ID: 0929.18003
DOI: 10.1016/S0022-4049(96)00150-8
author: Panagis Karazeris
published in: Journal of Pure and Applied Algebra
publication date: 2000-01-31T00:00:00Z
review text: 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).
Mathematics Subject Classification ID: 18B35
Mathematics Subject Classification ID: 03G25
Mathematics Subject Classification ID: 06F05
Mathematics Subject Classification ID: 16W80
zbMATH DE Number: 1308159
zbMATH Keywords: quantales
zbMATH Keywords: substructural logic
zbMATH Keywords: Gabriel topology
zbMATH Keywords: ring theoretic spectrum
zbMATH Keywords: local compactness
zbMATH Keywords: completeness theorem
reviewed by: K. I. Rosenthal
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1308159