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A number of models of linear logic are based on or closely related to linearalgebra, in the sense that morphisms are "matrices" over appropriatecoefficient sets. Examples include models based on coherence spaces, finitenessspaces and probabilistic coherence spaces, as well as the relational andweighted relational models. This paper introduces a unified framework based onmodule theory, making the linear algebraic aspect of the above models moreexplicit. Specifically we consider modules over Sigma-semirings $R$, which arering-like structures with partially-defined countable sums, and show thatmorphisms in the above models are actually $R$-linear maps in the standardalgebraic sense for appropriate $R$. An advantage of our algebraic treatment isthat the category of $R$-modules is locally presentable, from which it easilyfollows that this category becomes a model of intuitionistic linear logic withthe cofree exponential. We then discuss constructions of classical models andshow that the above-mentioned models are examples of our constructions.

Linear-Algebraic Models of Linear Logic as Categories of Modules over Sigma-Semirings