Proportional structures

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.