The paper aims at studying, in full generality, logics defined by imposing a variable inclusion condition on a given logic \(\vdash \). We prove that the description of the algebraic counterpart of the left variable inclusion companion of a given logic \(\vdash \) is related to the construction of Płonka sums of the matrix models of \(\vdash \). This observation allows to obtain a Hilbert-style axiomatization of the logics of left variable inclusion, to describe the structure of their reduced models, and to locate them in the Leibniz hierarchy.

中文翻译：

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左变量包含和矩阵的Płonka和的逻辑

本文旨在全面研究通过在给定逻辑\（\ vdash \）上施加变量包含条件而定义的逻辑。我们证明给定的逻辑左侧变量列入同伴的代数对方的描述\（\ vdash \）有关的矩阵模型的Płonka资金建设\（\ vdash \） 。该观察结果允许获得左变量包含逻辑的希尔伯特式公理化，以描述其简化模型的结构，并将其定位在莱布尼兹层次结构中。