Logic of left variable inclusion and Plonka sums of matrices

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.
Comment: to appear in Archive for Mathematical Logic