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

Quantale semantics of Lambek calculus with subexponential modalities