Estevan Muguerza, Asier
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Estevan Muguerza
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Asier
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Estadística, Informática y Matemáticas
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INARBE. Institute for Advanced Research in Business and Economics
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Publication Open Access Partial representations of orderings(World Scientific, 2018) Bosi, Gianni; Estevan Muguerza, Asier; Zuanon, Magali; Estadística, Informática y Matemáticas; Estatistika, Informatika eta MatematikaIn the present paper a new concept of representability is introduced, which can be applied to not total and also to intransitive relations (semiorders in particular). This idea tries to represent the orderings in the simplest manner, avoiding any unnecessary information. For this purpose, the new concept of representability is developed by means of partial functions, so that other common definitions of representability (i.e. (Richter-Peleg) multi-utility, Scott-Suppes representability, … ) are now particular cases in which the partial functions are actually functions. The paper also presents a collection of examples and propositions showing the advantages of this kind of representations, particularly in the case of partial orders and semiorders, as well as some results showing the connections between distinct kinds of representations.Publication Open Access Semiorders and continuous Scott–Suppes representations. Debreu’s Open Gap Lemma with a threshold(Elsevier, 2023) Estevan Muguerza, Asier; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika; Institute for Advanced Research in Business and Economics - INARBE; Universidad Pública de Navarra / Nafarroako Unibertsitate Publikoa, JIUPNA19-2022The problem of finding a utility function for a semiorder has been studied since 1956, when the notion of semiorder was introduced by Luce. But few results on continuity and no result like Debreu’s Open Gap Lemma, but for semiorders, was found. In the present paper, we characterize semiorders that accept a continuous representation (in the sense of Scott–Suppes). Two weaker theorems are also proved, which provide a programmable approach to Open Gap Lemma, yield a Debreu’s Lemma for semiorders, and enable us to remove the open-closed and closed-open gaps of a set of reals while keeping the threshold.Publication Open Access Searching for a Debreu’s open gap lemma for semiorders(Springer, 2020) Estevan Muguerza, Asier; Estatistika, Informatika eta Matematika; Institute for Advanced Materials and Mathematics - INAMAT2; Estadística, Informática y MatemáticasIn 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).