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A Note on Candeal and Induráin’s Semiorder Separability Condition

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Mathematical Topics on Representations of Ordered Structures and Utility Theory

Part of the book series: Studies in Systems, Decision and Control ((SSDC,volume 263))

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Abstract

We show that the semiorder separability condition used by Candeal and Induráin in their characterization of semiorders having a strict representation with positive threshold can be factorized into two conditions. The first says that the trace of the semiorder must have a numerical representation. The second asserts that the number of “noses” in the semiorder must be finite or countably infinite. We discuss the interest of such a factorization.

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Notes

  1. 1.

    Earlier results in the general case include Abrisqueta et al. [1], Candeal et al. [10], Gensemer [21,22,23], Narens [30].

  2. 2.

    Indeed, in this semiorder, for all \(x \in \mathbb {R}\), the ordered pair \((x+1, x)\) is a nose, as defined in Definition 4. Hence, we have an uncountable number of noses, while the existence of a strict numerical representation implies that the number of noses must be finite or countably infinite. See Remark 2.

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Acknowledgements

We would like to thank two referees for their useful comments on an earlier draft of this text.

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Authors and Affiliations

  1. LAMSADE, UMR 7243, CNRS, Université Paris-Dauphine, PSL Research University, 75016, Paris, France

    Denis Bouyssou

  2. Université de Mons, rue de Houdain 9, 7000, Mons, Belgium

    Marc Pirlot

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  1. Denis Bouyssou
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Correspondence to Denis Bouyssou .

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Editors and Affiliations

  1. Dipartimento di Scienze Economiche, Aziendali, Matematiche e Statistiche, Università degli Studi di Trieste, Trieste, Italy

    Gianni Bosi

  2. Departamento de Estadística, Informática y Matemáticas, Universidad Pública de Navarra, Pamplona, Navarra, Spain

    María J. Campión

  3. Facultad de Economía y Empresa, Departamento de Análisis Económico, Universidad de Zaragoza, Zaragoza, Spain

    Juan C. Candeal

  4. Departamento de Estadística, Informática y Matemáticas, Universidad Pública de Navarra, Pamplona, Spain

    Esteban Indurain

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Bouyssou, D., Pirlot, M. (2020). A Note on Candeal and Induráin’s Semiorder Separability Condition. In: Bosi, G., Campión, M., Candeal, J., Indurain, E. (eds) Mathematical Topics on Representations of Ordered Structures and Utility Theory. Studies in Systems, Decision and Control, vol 263. Springer, Cham. https://doi.org/10.1007/978-3-030-34226-5_6

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