Monads are commonplace in computer science, and can be composed using Beck's distributive laws. Unfortunately, finding distributive laws can be extremely difficult and error-prone. The literature contains some general principles for constructing distributive laws. However, until now there have been no such techniques for establishing when no distributive law exists. We present three families of theorems for showing when there can be no distributive law between two monads. The first widely generalizes a counterexample attributed to Plotkin. It covers all the previous known no-go results for specific pairs of monads, and includes many new results. The second and third families are entirely novel, encompassing various new practical situations. For example, they negatively resolve the open question of whether the list monad distributes over itself, reveal a previously unobserved error in the literature, and confirm a conjecture made by Beck himself in his first paper on distributive laws.

中文翻译：

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分配律的禁行定理

单子在计算机科学中很常见，可以使用贝克的分配定律来组成。不幸的是，找到分配律可能极其困难且容易出错。文献包含一些构建分配律的一般原则。然而，直到现在还没有这样的技术来确定不存在分配法时的情况。我们提出了三个定理系列，用于说明两个 monad 之间何时不存在分配律。第一个广泛概括了一个归因于 Plotkin 的反例。它涵盖了所有先前已知的特定 monad 对的 no-go 结果，并包括许多新结果。第二和第三家庭是全新的，包含各种新的实际情况。例如，