It is always possible to associate with an arbitrary propositional logic L, two substitution-invariant consequence relations which satisfy, respectively, a left and a right variable inclusion constraint and called the left and the right variable inclusion companion of L. For instance, the left and the right variable inclusion companions of classical (propositional) logic are, respectively, paraconsistent weak Kleene logic (PWK for short), and Bochvar logic. One of the main result of ours states that the algebraic counterpart of PWK is the class of Płonka sum of Boolean algebras. This observation led us to investigate the relations between left and right variable inclusion companions and Płonka sums in full generality. Our study is carried on in the conceptual framework of abstract algebraic logic.
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