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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.
Very likely my bad due to a false recollection of their argument, thanks for giving it a reality check! Hopefully, I got at least the substance right, namely, that product pregroup grammars land you in type 0. I’ll attend to the paragraph as soon as I find a minute. Feel free to revise concerning the Kracht-Kobele paper as you have a better overview of the relevant literature hence how the pieces fit together.
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