Lambek pregroups are Frobenius spiders in preorders

TL;DR

This paper reveals that Lambek pregroups can be viewed as Frobenius spiders within preorders, providing a new categorical perspective that links linguistics, algebra, and computational applications.

Contribution

It characterizes pregroups as pointed Frobenius spiders in preordered relations and shows preordered spider algebras as unions of pregroups, extending existing algebraic frameworks.

Findings

Pregroups are characterized as pointed spiders in preordered relations.

Preordered spider algebras can be expressed as unions of pregroups.

New categorical insights suggest applications in machine learning and data analysis.

Abstract

"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.