Quantale semantics of Lambek calculus with subexponential modalities

TL;DR

This paper develops a new semantic framework for the Lambek calculus with subexponential modalities using quantale semantics, establishing completeness and representation theorems.

Contribution

It introduces a novel interpretation of subexponential modalities via quantic conuclei and proves the calculus's completeness and relational completeness within this framework.

Findings

Lambek calculus with subexponentials is complete w.r.t. quantales with quantic conuclei.

A representation theorem for quantales with quantic conuclei is established.

The calculus is shown to be relationally complete.

Abstract

In this paper, we consider the polymodal version of Lambek calculus with subexponential modalities initially introduced by Kanovich, Kuznetsov, Nigam, and Scedrov and its quantale semantics. In our approach, subexponential modalities have an interpretation in terms of quantic conuclei. We show that this extension of Lambek calculus is complete w.r.t quantales with quantic conuclei. Also, we prove a representation theorem for quantales with quantic conuclei and show that Lambek calculus with subexponentials is relationally complete. Finally, we extend this representation theorem to the category of quantales with quantic conuclei.