| Summary: | 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.
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