Algebraizable Logics and a functorial encoding of its morphisms

TL;DR

This paper explores the categorical relationships between propositional logics and their structures, focusing on algebraizable logics and functorial encodings of their morphisms to advance the representation theory of general logics.

Contribution

It introduces a functorial encoding of morphisms between algebraizable logics, linking logic translations to functors on associated quasi-varieties, enhancing categorical logic analysis.

Findings

Morphisms between algebraizable logics can be fully encoded by specific functors.

The categorical framework facilitates the development of a representation theory for general logics.

The results connect logic translations with functorial structures, enabling new algebraic approaches.

Abstract

The present work presents some results about the categorial relation between logics and its categories of structures. A (propositional, finitary) logic is a pair given by a signature and Tarskian consequence relation on its formula algebra. The logics are the objects in our categories of logics; the morphisms are certain signature morphisms that are translations between logics (\cite{AFLM1},\cite{AFLM2},\cite{AFLM3} \cite{FC}). Morphisms between algebraizable logics (\cite{BP}) are translations that preserves algebraizing pairs (\cite{MaMe}): they can be completely encoded by certain functors defined on the quasi-variety canonically associated to the algebraizable logics. This kind of results will be useful in the development of a categorial approach to the representation theory of general logics (\cite{MaPi1}, \cite{MaPi2}, \cite{AJMP}).