Proportoids

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 -- 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.