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.

翻译：我们考虑将Lambek演算扩展交运算及显式零元和单位元常量后的关系语义（R-模型）。针对其不含常量且禁止空前件的变体，Andreka与Mikulas（1994）证明了强完备性。我们证明该性质在取消此限制后失效，但另一方面，针对非常量标准解释的情形，我们证明了弱完备性。对于标准解释，即使弱完备性亦不成立。该弱完备性结果可推广至无穷情形，即所谓迭代除法（克莱尼星号与除法的复合）。此外，我们还证明了无乘积片段的强完备性结果。