Logic of left variable inclusion and Plonka sums of matrices

TL;DR

This paper explores logics constrained by variable inclusion, showing their algebraic structure via Plonka sums of matrices, and provides axiomatizations and model descriptions for these logics.

Contribution

It introduces a general framework for variable inclusion logics and links them to Plonka sums, offering new axiomatizations and structural insights.

Findings

Algebraic characterization of variable inclusion logics via Plonka sums

Hilbert-style axiomatizations for these logics

Structural description of reduced models

Abstract

The paper aims at studying, in full generality, logics defined by imposing a variable inclusion condition on a given logic $\vdash$. It turns out that the algebraic counterpart of the variable inclusion companion of a given logic $\vdash$ is obtained by constructing the Plonka sum of the matrix models of $\vdash$. This association allows to obtain a Hilbert-style axiomatization of the logics of variable inclusion and to describe the structure of their reduced models.