Semimodules and the (syntactically-)linear lambda calculus
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In a recent paper, the ℒ^𝒮-calculus has been defined. It is a proof-language for a significant fragment of intuitionistic linear logic. Its main feature is that the linearity properties can be expressed in its syntax, since it has interstitial logical rules whose proof-terms are a sum and a multiplication by scalar. The calculus is parametrized on the structure 𝒮. This structure was originally identified with the field of complex numbers, since the calculus is designed as a quantum lambda calculus. However, in this paper we show that a semiring is enough, and we provide a categorical semantics for this calculus in the category of cancellative semimodules over the given semiring. We prove the semantics to be sound and adequate.
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