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It should be pointed out (somewhere, sometime) the relationship between pregroups and residuated lattices, which appears in much older work of Lambek dating back at least to the 60’s. This work of Lambek was all of a piece, connecting module theory to proof theory to linguistics. I would like to add some of this history at some point.
Re #6: This blog post is saying something very similar, using a monoidal category for sentence derivations. “That is, a string $u \in V^\star$ is grammatical whenever there exists an arrow from the start symbol $s$ to $u$ in $\mathcal{C}_R$.” Morphisms in monoidal categories can be notated using string diagrams.
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