Unit representation of semiorders I: Countable sets
Bouyssou, Denis; Pirlot, Marc (2020), Unit representation of semiorders I: Countable sets, Journal of Mathematical Psychology, 103, 102566. 10.1016/j.jmp.2021.102566
TypeArticle accepté pour publication ou publié
Journal nameJournal of Mathematical Psychology
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Laboratoire d'analyse et modélisation de systèmes pour l'aide à la décision [LAMSADE]
Abstract (EN)This paper proposes a new proof of the existence of constant threshold representations of semiorders on countably infinite sets. The construction treats each indifference-connected component of the semiorder separately. It uses a partition of such an indifference-connected component into indifference classes. Each element in the indifference-connected component is mirrored, using a “ghost” element, into a reference indifference class that is weakly ordered. A numerical representation of this weak order is used as the basis for the construction of the unit representation after an appropriate lifting operation. We apply the procedure to each indifference-connected component and assemble them adequately to obtain an overall unit representation.Our proof technique has several original features. It uses elementary tools and can be seen as the extension of a technique designed for the finite case, using a denumerable set of inductions. Moreover, it gives us much control on the representation that is built, so that it is, for example, easy to investigate its uniqueness. Finally, we show in a companion paper that our technique can be extended to the general (uncountable) case, almost without changes, through the addition of adequate order-denseness conditions.
Subjects / KeywordsSemiorder; Numerical representation; Constant threshold; Countable sets
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