Proportional algebras

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 introduces proportional algebras as algebras endowed with a 4-ary analogical proportion relation $a:b::c:d$ satisfying a suitable set of axioms. Functions preserving analogical proportions have already proven to be of practical interest in artificial intelligence and studying their mathematical properties is essential for understanding proportions. We therefore introduce proportional homomorphisms and their associated congruences and proportional functors, and show that they are closely related notions. In a broader sense, this paper is a further step towards a mathematical theory of analogical proportions and analogical reasoning in general.