Valuations and closure operators on finite lattices

Abstract

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.