Kwuida, Léonard; Schmidt, Stefan E.
(2011).
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Valuations and closure operators on finite lattices
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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 |