Kwuida, LéonardLéonardKwuidaSchmidt, Stefan E.Stefan E.Schmidt2024-11-192024-11-1920110166218X10.24451/arbor.12979https://doi.org/10.24451/arbor.1297910.1016/j.dam.2010.11.022https://arbor.bfh.ch/handle/arbor/31432Let L be a lattice. A function f : L → R (usually called evaluation) is submodular if f(x∧y)+f(x∨y) ≤ f(x)+f(y), supermodular if f(x∧y)+f(x∨y) ≥ f(x)+f(y), and modular if it is both submodular and supermodular. Modular functions on a finite lattice form a finite dimensional vector space. For finite distributive lattices, we compute this (modular) dimension. This turns out to be another characterization of distributivity (Theorem 3.9). We also present a correspondence between isotone submodular evaluations and closure operators on finite lattices (Theorem 5.5). This interplay between closure operators and evaluations should be understood as building a bridge between qualitative and quantitative data analysisenGeneralized measures on finite lattices Valuations Modular dimension Closure and kernel operators Qualitative data analysis Quantitative data analysisarticle