## General affine adjunctions, Nullstellensätze, and dualities

At last, we have finished and submitted our paper on “General affine adjunctions, Nullstellensätze, and dualities” co-authored with Olivia Caramello and Vincenzo Marra.

Abstract. We introduce and investigate a category-theoretic abstraction of the standard “system-solution” adjunction in affine algebraic geometry. We then look further into these geometric adjunctions at different levels of generality, from syntactic categories to (possibly infinitary) equational classes of algebras. In doing so, we discuss the relationships between the dualities induced by our framework and the well-established theory of concrete dual adjunctions. In the context of general algebra we prove an analogue of Hilbert’s Nullstellensatz, thereby achieving a complete characterisation of the fixed points on the algebraic side of the adjunction.

The preprint is available on arXiv.  We made another preprint available some years ago(!), but the manuscript has changed in many respects.  The main differences between the two versions on arXiv are the following:

1. The comparison with the existing literature is now more thorough.
2. The categories R and D are now taken directly without passing through the quotient categories. In our opinion, this is cleaner and, as a consequence, it is now clearer what are the minimal assumption on the triplet I: T -> S.
3. There is now a section studying the issue of concreteness of the adjunction and comparing with the theory of concrete adjunction.

## 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

## MV-algebras, infinite dimensional polyhedra, and natural dualities

Leo and I have just finished our paper on the connection between natural dualities and the duality between semisimple MV-algebras and compact Hausdorff spaces with definable maps. Actually, we provide a description of definable maps that is intrinsically geometric. In addition, we give some applications to semisimple tensor products, strongly semisimple and polyhedral MV-algebras.

## 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.