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The Möbius transform on symmetric ordered structures and its application to capacities on finite sets

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  • Michel Grabisch

    (DECISION - LIP6 - Laboratoire d'Informatique de Paris 6 - UPMC - Université Pierre et Marie Curie - Paris 6 - CNRS - Centre National de la Recherche Scientifique)

Abstract

Considering a linearly ordered set, we introduce its symmetric version, and endow it with two operations extending supremum and infimum, so as to obtain an algebraic structure close to a commutative ring. We show that imposing symmetry necessarily entails non associativity, hence computing rules are defined in order to deal with non associativity. We study in details computing rules, which we endow with a partial order. This permits to find solutions to the inversion formula underlying the Möbius transform. Then we apply these results to the case of capacities, a notion from decision theory which corresponds, in the language of ordered sets, to order preserving mappings, preserving also top and bottom. In this case, the solution of the inversion formula is called the Möbius transform of the capacity. Properties and examples of Möbius transform of sup-preserving and inf-preserving capacities are given.

Suggested Citation

  • Michel Grabisch, 2004. "The Möbius transform on symmetric ordered structures and its application to capacities on finite sets," Post-Print hal-00188158, HAL.
    • Handle: RePEc:hal:journl:hal-00188158
    DOI: 10.1016/j.disc.2004.05.013
    Note: View the original document on HAL open archive server: https://hal.science/hal-00188158
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    Cited by:

    1. Michel Grabisch & Christophe Labreuche, 2010. "A decade of application of the Choquet and Sugeno integrals in multi-criteria decision aid," Annals of Operations Research, Springer, vol. 175(1), pages 247-286, March.
    2. Miguel Couceiro & Michel Grabisch, 2013. "On the poset of computation rules for nonassociative calculus," PSE-Ecole d'économie de Paris (Postprint) hal-00787750, HAL.
    3. Michel Grabisch & Bernard de Baets & Janos Fodor, 2004. "The quest for rings on bipolar scales," Post-Print hal-00271217, HAL.
    4. Michel Grabisch, 2003. "The Symmetric Sugeno Integral," Post-Print hal-00272084, HAL.
    5. Dieter Denneberg & Michel Grabisch, 2004. "Measure and integral with purely ordinal scales," Post-Print hal-00272078, HAL.

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