Canonical formulas for k-potent commutative, integral, residuated lattices

Canonical formulas are a powerful tool for studying intuitionistic and modal logics. Actually, they provide a uniform and semantic way to axiomatise all extensions of intuitionistic logic and all modal logics above K4. Although the method originally hinged on the relational semantics of those logics, recently it has been completely recast in algebraic terms. In this new perspective canonical formulas are built from a finite subdirectly irreducible algebra by describing completely the behaviour of some operations and only partially the behaviour of some others. In this paper we export the machinery of canonical formulas to substructural logics by introducing canonical formulas for $k$-potent, commutative, integral, residuated lattices ($k$-$\mathsf{CIRL}$). We show that any subvariety of $k$-$\mathsf{CIRL}$ is axiomatised by canonical formulas. The paper ends with some applications and examples.Comment: Some typo corrected and additional comments adde

Goldblatt-Thomason Theorems for Modal Intuitionistic Logics

We prove Goldblatt-Thomason theorems for frames and models of a wide variety of modal intuitionistic logics, including ones studied by Wolter and Zakharyaschev, Goldblatt, Fischer Servi, and Plotkin and Sterling. We use the framework of dialgebraic logic to describe most of these logics and derive results in a uniform way