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 contributes to the mathematical foundations of analogical proportions in the axiomatic tradition as initiated by Yves Lepage two decades ago. For this we consider proportional structures as sets endowed with a 4-ary analogical proportion relation $a:b::c:d$ satisfying a suitable set of axioms and study different kinds of proportion-preserving mappings and relations and their properties. In a broader sense, this paper is a further step towards a mathematical theory of analogical proportions.

翻译：类比比例是"a对于b相当于c对于d"这一形式的表达式，是类比推理的核心，而类比推理本身又是人工智能的核心。本文延续了Yves Lepage二十年前开创的公理化传统，致力于为类比比例的数学基础做出贡献。为此，我们将比例结构视为具有四元类比比例关系a:b::c:d的集合，该关系满足一组适当的公理，并研究了不同类型的保比例映射与关系及其性质。从更广泛的意义上讲，本文是向类比比例的数学理论迈进的又一步。