Lambek pregroups are Frobenius spiders in preorders
This provides a foundational link between syntax and semantics in natural language processing, potentially enabling new applications in machine learning and data analysis.
The paper shows that Lambek pregroups, a linguistic structure, can be characterized as Frobenius spiders in preorders, and vice versa, extending known results from relational algebras to preordered settings.
"Spider" is a nickname of special Frobenius algebras, a fundamental structure from mathematics, physics, and computer science. Pregroups are a fundamental structure from linguistics. Pregroups and spiders have been used together in natural language processing: one for syntax, the other for semantics. It turns out that pregroups themselves can be characterized as pointed spiders in the category of preordered relations, where they naturally arise from grammars. The other way around, preordered spider algebras in general can be characterized as unions of pregroups. This extends the characterization of relational spider algebras as disjoint unions of groups. The compositional framework that emerged with the results suggests new ways to understand and apply the basis structures in machine learning and data analysis.