Valuations and closure operators on finite lattices

Authors:
Léonard Kwuida;Stefan E. Schmidt
Affiliations:
Zurich University of Applied Sciences, School of Engineering, Center of Applied Mathematics and Physics, Technikumstrasse 9, CH-8401 Winterthur, Switzerland;Technische Universität Dresden, Institut für Algebra, D-01062 Dresden, Germany
Venue:
Discrete Applied Mathematics
Year:
2011

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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.