Analogical reasoning is at the core of human and artificial intelligence and creativity. Analogical proportions are expressions of the form ``$a$ is to $b$ what $c$ is to $d$'' which are at the center of analogical reasoning which itself is at the core of artificial intelligence with numerous applications. 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 and studying their mathematical properties is essential for understanding proportions. We therefore introduce proportional homomorphisms (and their associated congruences) and functors and show that they are closely related notions. This provides us with mathematical tools for transferring knowledge across different domains which is crucial for future AI-systems. In a broader sense, this paper is a further step towards a mathematical theory of analogical reasoning.

翻译：类比推理是人类与人工智能及创造力的核心。类比比例是形如"$a$之于$b$如同$c$之于$d$"的表达，位于类比推理的中心，而类比推理本身又是人工智能的核心，具有众多应用。本文引入比例代数，将其定义为配备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被证明具有实际价值，而研究其数学性质对于理解比例至关重要。为此，我们引入比例同态（及其相关同余）和函子，并证明这些概念密切相关。这为我们提供了跨领域知识迁移的数学工具，这对未来人工智能系统至关重要。从更广泛的意义上讲，本文是迈向类比推理数学理论的又一步。