证明了在特定条件下，雅卡尔距离在任意格上满足三角不等式。
On the Triangle Inequality for the Jaccard Distance in Arbitrary Lattices
- 基于正性、单调性和模性，推广了雅卡尔距离的三角不等式
- 在相对补全分配格中，需满足超模和对数超模条件
- 适用于量子信息、形式概念分析等领域的度量计算
本文提出关于格上雅卡尔距离推广的新理论结果。当赋值严格为正、单调且为模时，雅卡尔距离在任意格上均满足三角不等式，推广了以往依赖分配性的结论。在相对补全分配格中，若赋值为正、单调、超模且对数超模，则三角不等式依然成立。我们还将对称差形式的雅卡尔距离拓展至分段补全分配格。进一步证明，超模性是标准广义雅卡尔距离作为有效度量的必要条件。最后，讨论了放宽结构约束在量子信息理论、形式概念分析及机器学习中的实际价值，并指出若干开放数学问题。
原文摘要 · Abstract (English)
This paper presents new theoretical results on generalizing the Jaccard distance for lattices and real valuations. We demonstrate that when the valuation is strictly positive, monotone, and modular, the Jaccard distance satisfies the triangle inequality on arbitrary lattices, effectively generalizing earlier results that depended heavily on distributivity. Moving to relatively complemented distributive lattices (which safely drop the requirement for the global bounds found in Boolean algebras), we prove the triangle inequality holds as long as the valuation is positive, monotone, supermodular, and $\log$-submodular. Additionally, we adapt the symmetric-difference Jaccard formulation for submodular valuations to sectionally complemented distributive lattices. Shifting to necessary conditions, we prove that supermodularity is a strict requirement for the standard generalized Jaccard distance to operate as a valid metric. Finally, we map the practical value of relaxing these structural constraints to computational fields like quantum information theory, formal concept analysis, and machine learning, closing with a brief look at open mathematical problems.
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