Valuations and closure operators on finite lattices

Let 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 analysis