Mp- and purified residuated lattices

Abstract

This paper applies a combination of algebraic and topological methods to obtain new and structural results on mp- and purified residuated lattices. It is demonstrated that mp-residuated lattices are strongly tied up with the dual hull-kernel topology. Mainly, it is shown that a residuated lattice is mp if and only if its minimal prime spectrum, equipped with the dual hull-kernel topology, is Hausdorff if and only if its prime spectrum, fitted with the dual hull-kernel topology, is normal. For a residuated lattice \({\mathfrak {A}}\) and a subalgebra \({\mathfrak {S}}\), the notion of disjunctive \({\mathfrak {S}}\)-regular and \({\mathfrak {S}}\)-mp-residuated lattices are introduced and investigated. It is shown that a residuated lattice \({\mathfrak {A}}\) is purified if and only if it is \(\beta ({\mathfrak {A}})\)-mp if and only if it is disjunctive \(\beta ({\mathfrak {A}})\)-regular, where \(\beta ({\mathfrak {A}})\) is the set of complemented elements of \({\mathfrak {A}}\). Some topological characterizations for purified residuated lattices are extracted and proved that a residuated lattice \({\mathfrak {A}}\) is purified if and only if its minimal prime spectrum, equipped with the dual hull-kernel topology, is a profinite space. To support mentioned concepts, we give place to some examples.