Proportional algebras

Analogical reasoning is at the core of human and artificial intelligence and creativity. Analogical proportions are expressions of the form ``$a$ is to $b$ what $c$ is to $d$'' which are at the center of analogical reasoning which itself is at the core of artificial intelligence with numerous applications. 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 and studying their mathematical properties is essential for understanding proportions. We therefore introduce proportional homomorphisms (and their associated congruences) and functors and show that they are closely related notions. This provides us with mathematical tools for transferring knowledge across different domains which is crucial for future AI-systems. In a broader sense, this paper is a further step towards a mathematical theory of analogical reasoning.