Flat topology on prime, maximal and minimal prime spectra of quantales

TL;DR

This paper extends topological characterizations of prime spectra from rings to quantales, providing new insights into their structure using flat and patch topologies, applicable to various algebraic systems.

Contribution

It generalizes topological results from ring theory to the broader context of quantales, introducing new characterizations of special classes of quantales via flat and patch topologies.

Findings

Characterization of hyperarchimedean quantales using topologies

Identification of normal and B-normal quantales through topological methods

Application of results to diverse algebraic structures like lattices and MV-algebras

Abstract

Several topologies can be defined on the prime, the maximal and the minimal prime spectra of a commutative ring; among them, we mention the Zariski topology, the patch topology and the flat topology. By using these topologies, Tarizadeh and Aghajani obtained recently new characterizations of various classes of rings: Gelfand rings, clean rings, absolutely flat rings, $mp$ - rings,etc. The aim of this paper is to generalize some of their results to quantales, structures that constitute a good abstractization for lattices of ideals, filters and congruences. We shall study the flat and the patch topologies on the prime, the maximal and the minimal prime spectra of a coherent quantale. By using these two topologies one obtains new characterization theorems for hyperarchimedean quantales, normal quantales, B-normal quantales, $mp$ - quantales and $PF$ - quantales. The general results can be applied to several concrete algebras: commutative rings, bounded distributive lattices, MV-algebras, BL-algebras, residuated lattices, commutative unital $l$ - groups, etc.