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