Syntactic Concept Lattice Models for Infinitary Action Logic

We introduce models for infinitary action logic, i.e., the infinitary extension of multiplicative-additive Lambek calculus with the Kleene star, on syntactic concept lattices. This semantics is a variant of language semantics, which is in a sense more natural from the linguistic point of view. Extending the argument of Wurm (2017), we prove completeness for the whole infinitary action logic, while standard language models enjoy completeness only for small fragments of this system. As a corollary, we obtain completeness of infinitary action logic w.r.t. action lattices which are complete in the lattice-theoretic sense.