Valuations and closure operators on finite lattices

Kwuida, Léonard; Schmidt, Stefan E. (2011). Valuations and closure operators on finite lattices Discrete Applied Mathematics, 159(10), pp. 990-1001. 10.1016/j.dam.2010.11.022

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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

Item Type:	Journal Article (Original Article)
Division/Institute:	Business School > Business Foundations and Methods
Name:	Kwuida, Léonard0000-0002-9811-0747 and Schmidt, Stefan E.
ISSN:	0166218X
Language:	English
Submitter:	Léonard Kwuida
Date Deposited:	01 Oct 2020 14:18
Last Modified:	06 Oct 2021 02:18
Publisher DOI:	10.1016/j.dam.2010.11.022
Uncontrolled Keywords:	Generalized measures on finite lattices Valuations Modular dimension Closure and kernel operators Qualitative data analysis Quantitative data analysis
ARBOR DOI:	10.24451/arbor.12979
URI:	https://arbor.bfh.ch/id/eprint/12979