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：我们考虑了扩展了交运算以及显式零和单位常数的 Lambek 演算的关系语义（R-模型）。对于其不带常数且限制空前件的变体，Andreka 和 Mikulas (1994) 证明了强完备性。我们指出，在没有这一限制的情况下，强完备性不成立；但另一方面，对于常数的非标准解释，我们证明了弱完备性。对于标准解释，即使弱完备性也不成立。该弱完备性结果可推广到无穷设定中，即所谓的迭代除法（除法下的 Kleene 星）。我们还证明了不含乘积片段的强完备性结果。