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    Searching for a Debreu’s open gap lemma for semiorders

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    2022-01-24
    Date
    2020
    Author
    Estevan Muguerza, Asier Upna
    Version
    Acceso embargado / Sarbidea bahitua dago
    xmlui.dri2xhtml.METS-1.0.item-type
    Capítulo de libro / Liburuen kapitulua
    Version
    Versión aceptada / Onetsi den bertsioa
    Project Identifier
    ES/1PE/MTM2015-63608-P 
    ES/1PE/ECO2015-65031 
    Impact
     
     
     
    10.1007/978-3-030-34226-5_5
     
     
     
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    Abstract
    In 1956 R. D. Luce introduced the notion of a semiorder to deal with indifference relations in the representation of a preference. During several years the problem of finding a utility function was studied until a representability characterization was found. However, there was almost no results on the continuity of the representation. A similar result to Debreu’s Lemma, but for semiorders was nev ... [++]
    In 1956 R. D. Luce introduced the notion of a semiorder to deal with indifference relations in the representation of a preference. During several years the problem of finding a utility function was studied until a representability characterization was found. However, there was almost no results on the continuity of the representation. A similar result to Debreu’s Lemma, but for semiorders was never achieved. In the present paper we propose a characterization for the existence of a continuous representation (in the sense of Scott-Suppes) for bounded semiorders. As a matter of fact, the weaker but more manageable concept of ε-continuity is properly introduced for semiorders. As a consequence of this study, a version of the Debreu’s Open Gap Lemma is presented (but now for the case of semiorders) just as a conjecture, which would allow to remove the open-closed and closed-open gaps of a subset S ⊆ R, but now keeping the constant threshold, so that x + 1 < y if and only if g(x) + 1 < g(y) (x, y ∈ S). [--]
    Subject
    Semiorders, Debreu’s Open Gap Lemma
     
    Publisher
    Springer
    Published in
    Bosi, G., Campión, M., Candeal, J., Indurain, E. (eds.) Mathematical Topics on Representations of Ordered Structures and Utility Theory. Springer: Cham, 2020, pp. 109-128. ISBN 978-3-030-34225-8.
    Departament
    Universidad Pública de Navarra. Departamento de Estadística, Informática y Matemáticas / Nafarroako Unibertsitate Publikoa. Estatistika, Informatika eta Matematika Saila / Universidad Pública de Navarra / Nafarroako Unibertsitate Publikoa. InaMat - Institute for Advanced Materials
     
    Publisher version
    https://doi.org/10.1007/978-3-030-34226-5_5
    URI
    https://hdl.handle.net/2454/36989
    Sponsorship
    The author acknowledges financial support from the Ministry of Economy and Competitiveness of Spain under grants MTM2015-63608-P and ECO2015-65031.
    Appears in Collections
    • Libros y capítulos de libros DEIM - EIMS Liburuak eta liburuen kapituluak [2]
    • Libros y capítulos de libros - Liburuak eta liburuen kapituluak [64]
    • Libros y capítulos de libros InaMat - InaMat liburuak eta liburuen kapituluak [1]
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