Detecting and exploiting similarities between seemingly distant objects is without doubt an important human ability. This paper develops \textit{from the ground up} an abstract algebraic and qualitative notion of similarity based on the observation that sets of generalizations encode important properties of elements. We show that similarity defined in this way has appealing mathematical properties. As we construct our notion of similarity from first principles using only elementary concepts of universal algebra, to convince the reader of its plausibility, we show that it can model fundamental relations occurring in mathematics and be naturally embedded into first-order logic via model-theoretic types. Finally, we sketch some potential applications to theoretical computer science and artificial intelligence.

翻译：检测并利用看似遥远对象之间的相似性无疑是人类的一项重要能力。本文从头开始发展了一种基于泛化的抽象代数与定性相似性概念，其核心观察在于：对象的泛化集合能够编码元素的重要性质。我们证明，以此方式定义的相似性具有令人满意的数学性质。由于我们仅使用泛代数基本概念从第一性原理构建此相似性概念，为说服读者其合理性，我们展示了该模型能够刻画数学中基本关系，并可通过模型论类型自然嵌入一阶逻辑。最后，我们概述了其在理论计算机科学与人工智能领域的若干潜在应用。