We consider relational semantics (R-models) for the Lambek calculus extended with intersection and explicit constants for zero and unit. For its variant without constants and a restriction which disallows empty antecedents, Andreka and Mikulas (1994) prove strong completeness. We show that it fails without this restriction, but, on the other hand, prove weak completeness for non-standard interpretation of constants. For the standard interpretation, even weak completeness fails. The weak completeness result extends to an infinitary setting, for so-called iterative divisions (Kleene star under division). We also prove strong completeness results for product-free fragments.

@article{paperbot2163,
  title = {Relational Models for the Lambek Calculus with Intersection and Constants},
  author = {Stepan L. Kuznetsov},
  journal = {Logical Methods in Computer Science},
  volume = {Volume 19, Issue 4},
  year = {2023},
  doi = {10.46298/lmcs-19(4:32)2023}
}