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 A and a subalgebra S the notion of disjunctive S-regular and S-mp residuated lattices introduced and investigated. It is shown that a residuated lattice A is purified if and only if it is (A)-mp if and only if it is disjunctive (A)-regular, where (A) is the set of complemented elements of A. Some topological characterizations for purified residuated lattices are extracted and proved that a residuated lattice A is purified if and only if its minimal prime spectrum, equipped with the dual hull-kernel topology, is a profinite space. 1

Normal Paradistributive Latticoids

“…After that they constructed quotient of residuated lattices via stabilizer and studied its properties. Rasouli and Dehghani [6] investigated the notion of an mp-residuated lattice, and extracted their topological characterizations. Rasouli and Kondo [7], introduced and investigated the notion of n-normal residuated lattice, as a subclass of residuated lattices in which every prime filter contains at most n minimal prime filters.…”