We study an expansion of the Distributive Non-associative Lambek Calculus with conjugates of the Lambek product operator and residuals of those conjugates. The resulting logic is well-motivated, under-investigated and difficult to tackle. We prove completeness for some of its fragments and establish that it is decidable. Completeness of the logic is an open problem; some difficulties with applying the usual proof method are discussed.

中文翻译：

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带共轭的兰贝克微积分

我们使用 Lambek 乘积算子的共轭和这些共轭的残差研究了分布式非关联兰贝克微积分的扩展。由此产生的逻辑动机良好，调查不足且难以解决。我们证明了它的一些片段的完整性并确定它是可判定的。逻辑的完整性是一个悬而未决的问题；讨论了应用通常的证明方法的一些困难。