## Join-completions of ordered algebras

In this paper, coauthored with José Gil-Férez, Constantine Tsinakis, and Hongjun Zhou, we present a systematic study of join-extensions and join-completions of ordered algebras, which naturally leads to a refined and simplified treatment of fundamental results and constructions in the theory of ordered structures ranging from properties of the Dedekind-MacNeille completion to the proof of the finite embeddability property for a number of varieties of ordered algebras.

ArXiv preprint

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

Nick, Nick and I have finished the paper about canonical formulas for $$k$$-potent residuated lattices.  A (incomplete) presentation of the paper can be found here.  The paper can be downloaded here

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 ($$k$$-????). We show that any subvariety of $$k$$-???? is axiomatised by canonical formulas. The paper ends with some applications and examples.