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.