Analogical proportions are expressions of the form ``$a$ is to $b$ what $c$ is to $d$'' at the core of analogical reasoning. This paper contributes to the mathematical foundations of analogical proportions in the axiomatic tradition as initiated -- in the tradition of the ancient Greeks -- by Yves Lepage two decades ago. More precisely, we first introduce the name ``proportoid'' for sets endowed with a 4-ary analogical proportion relation satisfying a suitable set of axioms. We then study study different kinds of proportion-preserving mappings and relations and their properties. Formally, we define homomorphisms of proportoids as mappings $\mathsf H$ satisfying $a:b::c:d$ iff $\mathsf Ha:\mathsf Hb::\mathsf Hc:\mathsf Hd$ for all elements and show that their kernel is a congruence. Moreover, we introduce (proportional) analogies as mappings $\mathsf A$ satisfying $a:b::\mathsf Aa:\mathsf Ab$ for all elements $a$ and $b$ in the source domain and show how to compute partial analogies. We then introduce a number of useful relations between functions (including homomorphisms and analogies) on proportoids and study their properties. In a broader sense, this paper is a further step towards a mathematical theory of analogical proportions.

翻译：类比比例是形如“$a$之于$b$如同$c$之于$d$”的表达，它是类比推理的核心。本文延续了二十年前Yves Lepage在古希腊传统下开创的公理化传统，致力于为类比比例建立数学基础。具体而言，我们首先引入“比例体”这一名称，用于指代配备满足特定公理集的四元类比比例关系的集合。随后，我们研究了不同类型的比例保持映射与关系及其性质。形式上，我们将比例体的同态定义为满足$a:b::c:d$当且仅当$\mathsf Ha:\mathsf Hb::\mathsf Hc:\mathsf Hd$对所有元素成立的映射$\mathsf H$，并证明其核为同余关系。此外，我们引入（比例）类比映射$\mathsf A$，要求其满足$a:b::\mathsf Aa:\mathsf Ab$对定义域中所有元素$a$和$b$成立，并展示了如何计算部分类比映射。接着，我们建立了比例体上函数（包括同态与类比映射）之间的若干实用关系，并研究了它们的性质。从更广泛的意义上说，本文是朝着建立类比比例数学理论迈出的又一步。