On the Triangle Inequality for the Jaccard Distance in Arbitrary Lattices

This paper extends the Jaccard distance to arbitrary lattices and real valuations, proving that the triangle inequality holds when the valuation is strictly positive, monotone, and modular. It further shows that in relatively complemented distributive lattices the inequality remains valid under positive, monotone, supermodular, and log‑submodular valuations, and adapts the symmetric‑difference Jaccard formulation for submodular valuations in sectionally complemented distributive lattices. The authors also establish that supermodularity is a necessary condition for the generalized Jaccard distance to be a metric and discuss the implications of these relaxed structural constraints for fields such as quantum information theory, formal concept analysis, and machine learning.

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