一句话:在通用代数类比比例框架下研究布尔域比例,并证明其与文献中两种著名模型一致。
本文在作者近期提出的通用代数类比比例抽象框架中,考察由 0 和 1 构成的布尔域上的类比比例。研究目标是刻画布尔比例这一特殊情形,并将其与既有文献中的建模方式联系起来。结果表明,该框架下定义的布尔比例与文献中在不同背景下提出的两个著名模型相吻合。这意味着两个原本独立的布尔比例建模可以被统一到同一框架中,在数学上更具吸引力,也为该通用框架的稳健性与适用性提供了进一步证据。
The author has recently introduced an abstract algebraic framework of analogical proportions within the general setting of universal algebra. This paper studies analogical proportions in the boolean domain consisting of two elements 0 and 1 within his framework. It turns out that our notion of boolean proportions coincides with two prominent models from the literature in different settings. This means that we can capture two separate modellings of boolean proportions within a single framework which is mathematically appealing and provides further evidence for the robustness and applicability of the general framework.