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.

本文介绍了比例代数作为一种代数，并研究了保持比例关系的函数的性质，提供了数学工具，以便将知识传递到不同的领域，这对未来的人工智能系统至关重要。此外，本文是朝着比拟推理的数学理论迈出的更进一步的一步。