Lambek pregroups are Frobenius spiders in preorders
Dusko Pavlovic
TL;DR
The paper establishes a tight correspondence between Lambek pregroups and Frobenius spiders within the category of preordered relations, using a sequent-algebraic framework. It proves that pregroups are precisely pointed spiders and that Frobenius spider structures align with residuated monoids, while showing that every spider in preorders can be decomposed into a consistent union of pregroups (and vice versa). This unifies syntactic and semantic formalisms under a common sequent-algebraic lens and extends known characterizations from relational spiders over sets to preordered relations. The results have potential implications for linguistic models, data analysis, and machine learning by providing a compositional, algebraic lens that links grammar reductions to semantic Spider decompositions and their representations. The paper also outlines future directions, including Cayley-style representations, pregroup coverings, and deeper connections to DisCoCat and vector-space semantics within a unified framework.
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.