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arXiv · 2507.06007

A Proof Theory for Profinite Modal Algebras

Matteo De Berardinis·Silvio Ghilardi
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Abstract

In a previous paper, we showed that profinite $L$-algebras (where $L$ is a variety of modal algebras generated by its finite members) are monadic over $\mathbf{Set}$. This monadicity result suggests that profinite $L$-algebras could be presented as Lindenbaum algebras for propositional theories in infinitary versions of propositional modal calculi. In this paper we identify such calculi as modal enrichments of Maehara-Takeuti's infinitary extension of the sequent calculus $\mathbf{LK}$. We also investigate correspondences between syntactic properties of the calculi and regularity/exactness properties of the opposite category of profinite $L$-algebras.

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Matteo De Berardinis, Silvio Ghilardi. 2026-09-21. A Proof Theory for Profinite Modal Algebras. https://arxiv.org/abs/2507.06007

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