Analogical reasoning is the ability to detect parallels between two seemingly distant objects or situations, a fundamental human capacity used for example in commonsense reasoning, learning, and creativity which is believed by many researchers to be at the core of human and artificial general intelligence. Analogical proportions are expressions of the form ``$a$ is to $b$ what $c$ is to $d$'' at the core of analogical reasoning. The author has recently introduced an abstract algebraic framework of analogical proportions within the general setting of universal algebra. It is the purpose of this paper to further develop the mathematical theory of analogical proportions within that framework as motivated by the fact that it has already been successfully applied to logic program synthesis in artificial intelligence.

翻译：类比推理是一种能够识别两个看似遥远对象或情境之间相似性的能力，这是人类的一项基本认知功能，广泛应用于常识推理、学习与创造力中，并被许多研究者认为是人类与通用人工智能的核心所在。类比比例是类比推理的核心表达形式，其结构为“$a$ 之于 $b$ 如同 $c$ 之于 $d$”。作者近期在泛代数的一般框架下提出了类比比例的抽象代数理论。本文旨在该框架下进一步发展类比比例的数学理论，其动机在于该理论已在人工智能领域的逻辑程序合成中得到成功应用。