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Lambek演算扩展了交和显式零与单位常数后的关系语义（R-模型）。对于其无常数且禁止空前件的变体，Andreka和Mikulas（1994）证明了强完全性。我们证明，在没有此限制的情况下强完全性失败；但另一方面，对于常数的非标准解释，我们证明了弱完全性。对于标准解释，即使是弱完全性也不成立。弱完全性结果可推广到无限情形，即所谓迭代除法（除法下的Kleene星）。我们还证明了无积片段的强完全性结果。