Duality for Normal Lattice Expansions and Sorted, Residuated Frames with Relations

TL;DR

This paper presents a simplified and improved duality theory for lattices with quasioperators, using sorted, residuated frames and relations, which clarifies the algebraic and logical structures involved.

Contribution

It introduces a streamlined duality framework for lattices with quasioperators, incorporating section stability and redefining morphisms to preserve Galois stable sets, advancing the understanding of non-distributive logics.

Findings

Simplified axiomatization of frames using section stability.

Redefinition of morphisms to preserve Galois stable sets.

Representation of non-distributive logics as fragments of sorted, residuated (poly)modal logics.

Abstract

We revisit the problem of Stone duality for lattices with various quasioperators, first studied in [14], presenting a fresh duality result. The new result is an improvement over that of [14] in two important respects. First, the axiomatization of frames in [14] was rather cumbersome and it is now simplified, partly by incorporating Gehrke's proposal [8] of section stability for relations. Second, morphisms are redefined so as to preserve Galois stable (and co-stable) sets and we rely for this, partly again, on Goldblatt's [11] recently proposed definition of bounded morphisms for polarities, though we need to strengthen the definition in order to get a Stone duality result. In studying the dual algebraic structures associated to polarities with relations we demonstrate that stable/co-stable set operators result as the Galois closure of the restriction of classical (though sorted) image operators generated by the frame relations to Galois stable/co-stable sets. This provides a proof, at the representation level, that non-distributive logics can be viewed as fragments of sorted, residuated (poly)modal logics, a research direction initiated in [16,17].