Ideals in quantales, II

TL;DR

This paper explores the topological properties of ideals in commutative quantales, focusing on sobriety, quasi-compactness, spectrality, and connectedness, and introduces new concepts like strongly disconnected spaces.

Contribution

It extends the theory of ideals in quantales by analyzing their topological properties and establishing conditions for sobriety, spectrality, and connectedness of quantale spaces.

Findings

Quantale spaces of proper ideals are spectral as per Hochster's definition.

Strongly disconnected quantale spaces with all maximal ideals imply the existence of non-trivial idempotents.

A criterion for the connectedness of quantale spaces is provided.

Abstract

As a natural extension of the ongoing development of a theory of ideals in commutative quantales with an identity element, this article aims to study into the analysis of certain topological properties exhibited by distinguished classes of ideals. These ideals are equipped with quantale topologies. The primary objectives encompass characterizing quantale spaces exhibiting sobriety, examining several conditions pertaining to quasi-compactness, and demonstrating that quantale spaces comprised of proper ideals adhere to the spectral properties as defined by Hochster. We introduce the notion of a strongly disconnected spaces and show that for a quantale with zero Jacobson radical, strongly disconnected quantale spaces containing all maximal ideals of the quantale imply existence of non-trivial idempotent elements in the quantale. Additionally, a sufficient criterion for establishing the connectedness of a quantale space is presented. Finally, we discuss on continuous maps between quantale spaces.