Jaccard Distance Triangle Inequality Generalized for Arbitrary Lattices.

This research delves into the theoretical foundations of the Jaccard distance, extending its applicability to more general mathematical structures known as lattices, beyond the traditional Boolean algebras. The paper demonstrates that the triangle inequality, a fundamental property for any valid metric, holds for the Jaccard distance on arbitrary lattices when the valuation function is strictly positive, monotone, and modular. This significantly broadens previous results that relied on distributivity. Further, the study explores relatively complemented distributive lattices, proving the triangle inequality under conditions of positive, monotone, supermodular, and log-submodular valuations. It also adapts the symmetric-difference Jaccard formulation for submodular valuations to sectionally complemented distributive lattices. Crucially, the paper establishes that supermodularity is a necessary condition for the standard generalized Jaccard distance to function as a valid metric. The practical implications of these theoretical relaxations are discussed for fields such as quantum information theory, formal concept analysis, and machine learning.