MP- and Purified residuated lattices

Abstract In this paper, a combination of algebraic and topological methods is applied to obtain new and structural results on mp and purified residuated lattices. It is demonstrated that mp-residuated lattices strongly tied up with the dual hull-kernel topology. Especially, 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, equipped 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 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.