We investigate language interpretations of two extensions of the Lambek calculus: with additive conjunction and disjunction and with additive conjunction and the unit constant. For extensions with additive connectives, we show that conjunction and disjunction behave differently. Adding both of them leads to incompleteness due to the distributivity law. We show that with conjunction only no issues with distributivity arise. In contrast, there exists a corollary of the distributivity law in the language with disjunction only which is not derivable in the non-distributive system. Moreover, this difference keeps valid for systems with permutation and/or weakening structural rules, that is, intuitionistic linear and affine logics and affine multiplicative-additive Lambek calculus. For the extension of the Lambek calculus with the unit constant, we present a calculus which reflects natural algebraic properties of the empty word. We do not claim completeness for this calculus, but we prove undecidability for the whole range of systems extending this minimal calculus and sound w.r.t. language models. As a corollary, we show that in the language with the unit there exists a sequent that is true if all variables are interpreted by regular language, but not true in language models in general.

我们研究了 Lambek 演算的两个扩展的语言解释：加法合取和析取以及加法合取和单位常数。对于具有加法连接词的扩展，我们展示了合取和析取的行为不同。由于分配律，将它们两者相加会导致不完整性。我们表明，只有结合才不会出现分配问题。相反，只有析取的语言中存在分配律的推论，在非分配系统中是不可推导的。此外，这种差异对于具有置换和/或弱化结构规则的系统仍然有效，即直觉线性和仿射逻辑以及仿射乘加加法 Lambek 演算。对于具有单位常数的 Lambek 微积分的扩展，我们提出了一种微积分，它反映了空词的自然代数性质。我们不要求这个微积分的完整性，但我们证明了扩展这个最小微积分和健全的语言模型的整个系统范围的不可判定性。作为推论，我们表明，在具有单位的语言中，如果所有变量都由常规语言解释，则存在一个序列，但在一般语言模型中则不成立。