The Lambek calculus (a variant of intuitionistic linear logic initially introduced for mathematical linguistics) enjoys natural interpretations over the algebra of formal languages (L-models) and over the algebra of binary relations which are subsets of a given transitive relation (R-models). For both classes of models there are completeness theorems (Andreka and Mikulas [J. Logic Lang. Inf., 3, No. 1, 1-37 (1994)]; Pentus [Ann. Pure Appl. Logic, 75, Nos. 1/2, 179-213 (1995); Fund. Prikl. Mat., 5, No. 1, 193- 219 (1999)]). The operations of the Lambek calculus include product and two divisions, left and right. We consider an extension of the Lambek calculus with intersection and iteration (Kleene star). It is proved that this extension is incomplete both w.r.t. L-models and w.r.t. R-models. We introduce a restricted fragment, in which iteration is allowed only in denominators of division operations. For this fragment we prove completeness w.r.t. R-models. We also prove completeness w.r.t. L-models for the subsystem without product. Both results are strong completeness theorems, i.e., they establish equivalence between derivability from sets of hypotheses (finite or infinite) and semantic entailment from sets of hypotheses on a given class of models. Finally, we prove $$ {\Pi}_1^0 $$ -completeness of the algorithmic problem of derivability in the restricted fragment in question.

中文翻译：

---

带迭代和交运算的兰贝克微积分的一个受限片段

Lambek 演算（最初为数学语言学引入的直觉线性逻辑的一种变体）对形式语言（L 模型）的代数和作为给定传递关系（R 模型）的子集的二元关系的代数具有自然的解释. 对于这两类模型，都有完备性定理（Andreka 和 Mikulas [J. Logic Lang. Inf., 3, No. 1, 1-37 (1994)]; Pentus [Ann. Pure Appl. Logic, 75, Nos. 1 /2, 179-213 (1995); Fund. Prikl. Mat., 5, No. 1, 193-219 (1999)])。Lambek 演算的运算包括乘积和左和右两个除法。我们考虑具有交集和迭代的 Lambek 演算的扩展（Kleene 星）。事实证明，这个扩展对于 L 模型和 R 模型都是不完整的。我们引入一个受限片段，其中迭代只允许在除法运算的分母中。对于这个片段，我们证明了 R 模型的完整性。我们还证明了没有产品子系统的 L 模型的完整性。这两个结果都是强完备性定理，即它们建立了从假设集（有限或无限）的可推导性和从给定模型类的假设集的语义蕴涵之间的等价性。最后，我们证明了 $$ {\Pi}_1^0 $$ - 所讨论的受限片段中可推导性算法问题的完整性。它们建立了从假设集（有限或无限）的可推导性和从给定模型类别的假设集的语义蕴涵之间的等价性。最后，我们证明了 $$ {\Pi}_1^0 $$ - 所讨论的受限片段中可推导性算法问题的完整性。它们建立了从假设集（有限或无限）的可推导性和从给定模型类别的假设集的语义蕴涵之间的等价性。最后，我们证明了 $$ {\Pi}_1^0 $$ - 所讨论的受限片段中可推导性算法问题的完整性。