On syntactic concept lattice models for the Lambek calculus and infinitary action logic

The linguistic applications of the Lambek calculus suggest its semantics over algebras of formal languages. A straightforward approach to construct such semantics indeed yields a brilliant completeness theorem (Pentus 1995, Ann. Pure Appl. Logic, 75, 179–213). However, extending the calculus with extra operations ruins completeness. In order to mitigate this issue, Wurm (2017, J. Logic Lang. Inf., 26, 179–214) introduced a modification of this semantics, namely, models over syntactic concept lattices. We extend this semantics to the infinitary extension of the Lambek calculus with Kleene iteration (infinitary action logic), prove strong completeness and some interesting corollaries. We also discuss issues arising with constants—zero, unit, top—and provide some strengthenings of Wurm’s results towards including these constants into the systems involved.