Duality for Normal Lattice Expansions and Sorted, Residuated Frames with Relations
Chrysafis Hartonas
TL;DR
This work resolves Stone duality for lattices with quasioperators by developing sorted residuated frames and canonical frames that robustly represent normal lattice expansions ($NLE_\tau$). It simplifies frame axioms using Gehrke's section stability, extends morphisms to preserve all Galois-stable/co-stable structures, and derives operator semantics from frame relations via Galois closures. The authors establish a full Stone-type duality through a dual category ${\mathbf{SRF}}^*_{\tau}$ and functors ${\tt F}$ and ${\tt L}$ between ${\bf NLE}_\tau$ and ${\bf SRF}^*_\tau$, showing non-distributive logics as fragments of sorted residuated polymodal logics. This representation-theoretic bridge has potential to transfer questions about non-distributive logics into the well-developed, modular framework of sorted residuated modalities, offering new avenues for analysis and applications in logic and lattice theory.
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].