The pure spectrum of a residuated lattice

TL;DR

This paper explores the structure of pure filters in residuated lattices, establishing topological properties of their spectra and connecting pure spectra with classical spectra in special cases.

Contribution

It introduces the notion of pure filters in residuated lattices, investigates pure-prime filters, and proves new theorems relating pure spectra to known topological spaces.

Findings

Pure spectrum is a compact sober space.

Pure spectrum of Gelfand residuated lattice is Hausdorff.

Pure spectrum coincides with maximal spectrum in mp-residuated lattices.

Abstract

This paper studies a fascinating type of filter in residuated lattices, the so-called pure filters. A combination of algebraic and topological methods on the pure filters of a residuated lattice is applied to obtain some new structural results. The notion of purely-prime filters of a residuated lattice has been investigated, and a Cohen-type theorem has been obtained. It is shown that the pure spectrum of a residuated lattice is a compact sober space, and a Grothendieck-type theorem has been demonstrated. It is proved that the pure spectrum of a Gelfand residuated lattice is a Hausdorff space, and deduced that the pure spectrum of a Gelfand residuated lattice is homeomorphic to its usual maximal spectrum. Finally, the pure spectrum of an mp-residuated lattice is investigated and verified that a given residuated lattice is mp iff its minimal prime spectrum is equipped with the induced dual hull-kernel topology, and its pure spectrum is the same.