HMS-Duality for Residuated Lattices
Josef C L Doyle von Hoffmann

Abstract:
In this thesis we present a novel topological duality for not-necessarily-distributive residuated lattice ordered groupoids by modifying a recent duality for bounded lattices established by Bezhanishvili et al. (2024). Our duality establishes a natural connection between the algebraic semantics of substructural logics and the operational frame semantics originating in the work of Ono and Komori (1985), Humberstone (1987), and Došen (1989). This allow us to the further generalize the original completeness theorems for the operational semantics and to gain insight into the success of canonical model style proofs that were utilized. In particular we adapt a notion of persistence from Bezhanishvili et al. (2024) and show that the canonical model style proofs in Ono and Komori (1985), Humberstone (1987), and Došen (1989) can be explained by an analysis given in terms of algebraic completeness, topological duality, and the salient notion of persistence. We also explore the duality in its own right and obtain topological representations of the lattice of congruences and products of residuated lattices.