Relational models for the Lambek calculus with intersection and unit
Apparaît dans la collection : 19th International Conference on Relational and Algebraic Methods in Computer Science / 19ème Conférence internationale de méthodes relationnelles et algébriques en informatique
We consider the Lambek calculus extended with intersection (meet) operation. For its variant which does not allow empty antecedents, Andreka and Mikulas (1994) prove strong completeness w.r.t. relational models (R-models). Without the antecedent non-emptiness restriction, however, only weak completeness w.r.t. R-models (so-called square ones) holds (Mikulas 2015). Our goals are as follows. First, we extend the calculus with the unit constant, introduce a class of non-standard R-models for it, and prove completeness. This gives a simpler proof of Mikulas' result. Second, we prove that strong completeness does not hold. Third, we extend our weak completeness proof to the infinitary setting, to so-called iterative divisions (Kleene star under division).
