We axiomatically characterize the orderings of pairs of sets induced by several distances: Hamming, Jaccard, Sørensen-Dice and Overlap. We also axiomatically characterize these distances. Our axioms are properties describing how a distance changes when we perform elementary modifications of the sets, like adding one element to one of the sets, to both sets, swapping both sets, permuting some elements, etc.

翻译：我们公理化地刻画了由若干距离诱导的集合对序关系：汉明距离、杰卡德距离、索伦森-戴斯距离以及重叠距离。同时，我们也对这些距离本身进行了公理化刻画。所采用的公理描述了当对集合进行基本修改时距离如何变化，例如向其中一个集合添加一个元素、向两个集合添加元素、交换两个集合、置换某些元素等操作。