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).[-]