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The quest for rings on bipolar scales

Author

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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)

  • Bernard de Baets

    (UGENT - Universiteit Gent = Ghent University)

  • Janos Fodor

    (Szent Istvan University - Szent Istvan University)

Abstract

We consider the interval $]{-1},1[$ and intend to endow it with an algebraic structure like a ring. The motivation lies in decision making, where scales that are symmetric w.r.t.~$0$ are needed in order to represent a kind of symmetry in the behaviour of the decision maker. A former proposal due to Grabisch was based on maximum and minimum. In this paper, we propose to build our structure on t-conorms and t-norms, and we relate this construction to uninorms. We show that the only way to build a group is to use strict t-norms, and that there is no way to build a ring. Lastly, we show that the main result of this paper is connected to the theory of ordered Abelian groups.

Suggested Citation

  • Michel Grabisch & Bernard de Baets & Janos Fodor, 2004. "The quest for rings on bipolar scales," Post-Print hal-00271217, HAL.
    • Handle: RePEc:hal:journl:hal-00271217
    Note: View the original document on HAL open archive server: https://hal.science/hal-00271217
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    References listed on IDEAS

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    1. De Baets, Bernard, 1999. "Idempotent uninorms," European Journal of Operational Research, Elsevier, vol. 118(3), pages 631-642, November.
    2. Tversky, Amos & Kahneman, Daniel, 1992. "Advances in Prospect Theory: Cumulative Representation of Uncertainty," Journal of Risk and Uncertainty, Springer, vol. 5(4), pages 297-323, October.
    3. Michel Grabisch, 2003. "The Symmetric Sugeno Integral," Post-Print hal-00272084, HAL.
    4. Michel Grabisch, 2004. "The Möbius transform on symmetric ordered structures and its application to capacities on finite sets," Post-Print hal-00188158, HAL.
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    Cited by:

    1. Michel Grabisch, 2006. "Aggregation on bipolar scales," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-00187155, HAL.
    2. Miguel Couceiro & Michel Grabisch, 2013. "On the poset of computation rules for nonassociative calculus," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) hal-00787750, HAL.

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