Gelfand residuated lattices

TL;DR

This paper explores the structure of Gelfand residuated lattices using algebraic and topological methods, revealing their connection with the hull-kernel topology and introducing the class of soft residuated lattices.

Contribution

It provides new characterizations of Gelfand residuated lattices through topological properties, pure filters, and introduces the concept of soft residuated lattices.

Findings

Gelfand residuated lattices are characterized by the normality of their prime spectrum.

A residuated lattice is Gelfand iff its pure spectrum is homeomorphic to its maximal spectrum.

The class of soft residuated lattices is characterized as Gelfand and semisimple.

Abstract

In this paper, a combination of algebraic and topological methods is applied to obtain new and structural results on Gelfand residuated lattices. It is demonstrated that Gelfand's residuated lattices strongly tied up with the hull-kernel topology. Especially, it is shown that a residuated lattice is Gelfand if and only if its prime spectrum, equipped with the hull-kernel topology, is normal. The class of soft residuated lattices is introduced, and it is shown that a residuated lattice is soft if and only if it is Gelfand and semisimple. Gelfand residuated lattices are characterized using the pure part of filters. The relation between pure filters and radicals in a Gelfand residuated lattice is described. It is shown that a residuated lattice is Gelfand if and only if its pure spectrum is homeomorphic to its usual maximal spectrum. The pure filters of a Gelfand residuated lattice are characterized. Finally, it is proved that a residuated lattice is Gelfand if and only if the hull-kernel and the $\mathscr{D}$-topology coincide on the set of maximal filters.