Name
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. |
Year | DOI | Venue |
|---|---|---|
2011 | 10.1016/j.dam.2010.11.022 | Discrete Applied Mathematics |
Keywords | Field | DocType |
closure and kernel operators,closure operator,quantitative data analysis,finite lattice form,qualitative data analysis,finite lattice,generalized measures on finite lattices,finite distributive lattice,isotone submodular evaluation,modular function,modular dimension,finite dimensional vector space,valuations,vector space,data analysis,distributive lattice | Distributive property,Discrete mathematics,Combinatorics,Vector space,Closure operator,Lattice (order),Distributivity,Submodular set function,Operator (computer programming),Isotone,Mathematics | Journal |
Volume | Issue | ISSN |
159 | 10 | Discrete Applied Mathematics |
Citations | PageRank | References |
1 | 0.36 | 10 |
Authors | ||
2 | ||
Name | Order | Citations | PageRank |
|---|---|---|---|
| Léonard Kwuida | 1 | 55 | 16.25 |
| Stefan E. Schmidt | 2 | 20 | 7.34 |