Analogical proportions are expressions of the form "$a$ is to $b$ what $c$ is to $d$" at the core of analogical reasoning which itself is at the core of artificial intelligence. This paper introduces proportional algebras as algebras endowed with a 4-ary analogical proportion relation $a:b::c:d$ satisfying a suitable set of axioms. Functions preserving analogical proportions have already proven to be of practical interest in artificial intelligence and studying their mathematical properties is essential for understanding proportions. We therefore introduce proportional homomorphisms and their associated congruences and proportional functors, and show that they are closely related notions. In a broader sense, this paper is a further step towards a mathematical theory of analogical proportions and analogical reasoning in general.

翻译：类比比例是形如"a之于b如同c之于d"的表达，处于类比推理的核心地位，而类比推理本身又是人工智能的核心。本文引入比例代数作为具有满足适当公理集的四元类比比例关系 a:b::c:d 的代数结构。保持类比比例的函数已在人工智能中被证明具有实际价值，研究其数学性质对于理解比例至关重要。因此，我们引入比例同态及其相关联的余核与比例函子，并证明这些概念之间存在密切联系。从更广泛的意义上说，本文是向建立类比比例乃至类比推理的数学理论迈进的又一步。