Continuous Representability of Interval Orders: The Topological Compatibility Setting

Bosi, Gianni; Estevan, Asier; García, Javier Gutiérrez; Induráin, Esteban

doi:10.1142/s0218488515500142

Cited by 14 publications

(14 citation statements)

References 40 publications

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“…there exists r 1 > 0 such that ∑ +∞ k=1 δ 1 k ≤ 1 − r 1 ). We denote the family of bad gaps in I 1 corresponding to the representation u 2…”

Section: Furthermorementioning

confidence: 99%

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Searching for a Debreu’s Open Gap Lemma for Semiorders

Muguerza

2020

Mathematical Topics on Representations of Ordered Structures and Utility Theory

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The problem of finding a (continuous) utility function for a semiorder has been studied since in 1956 R.D. Luce introduced in Econometrica the notion. There was almost no results on the continuity of the representation. A similar result to Debreu's Lemma, but for semiorders, was never achieved. Recently, some necessary conditions for the existence of a continuous representation as well as some conjectures were presented by A. Estevan. In the present paper we prove these conjectures, achieving the desired version of Debreu's Open Gap Lemma for bounded semiorders. This result allows 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). Therefore, the continuous representation (in the sense of Scott-Suppes) of bounded semiorders is characterized. These results are achieved thanks to the key notion of ε-continuity, which generalizes the idea of continuity for semiorders.

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“…there exists r 1 > 0 such that ∑ +∞ k=1 δ 1 k ≤ 1 − r 1 ). We denote the family of bad gaps in I 1 corresponding to the representation u 2…”

Section: Furthermorementioning

confidence: 99%

“…The length of the biggest gap G 2 1 corresponds to the expansion of the gap G 2 of u 1 (X), thus, the length of the gap G 2 1 is l 2 =…”

Section: Furthermorementioning

confidence: 99%

Searching for a Debreu’s Open Gap Lemma for Semiorders

Muguerza

2020

Mathematical Topics on Representations of Ordered Structures and Utility Theory

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“…This is due to the fact that nothing has been said about the possibility of the existence of a pair (u, v) of continuous functions representing ≺ but such that either u does not represent * * or v does not represent * . In this direction, some further discussion can be seen in Section 3 in [129], where other (still partial) solutions to the problem of characterizing the continuous representability of interval orders have been achieved. These are based on topological conditions different from the ones involved in the main results launched in [46].…”

Section: Remark 26mentioning

confidence: 99%

A Survey on the Mathematical Foundations of Axiomatic Entropy: Representability and Orderings

Campión

Gómez‐Polo

Induráin

et al. 2018

Axioms

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Different abstract versions of entropy, encountered in science, are interpreted in the light of numerical representations of several ordered structures, as total-preorders, interval-orders and semiorders. Intransitivities, other aspects of entropy as competitive systems, additivity, etc., are also viewed in terms of representability of algebraic structures endowed with some compatible ordering. A particular attention is paid to the problem of the construction of an entropy function or their mathematical equivalents. Multidisciplinary comparisons to other similar frameworks are also discussed, pointing out the mathematical foundations.

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“…Anyway, notice that we can obtain some particular results on continuous realizable (generalized) biorders dealing with the traces of the (generalized) 7 In fact, it is a representable interval order, so it is well know that it is actually i.o.-separable (see [5,12]). 8 Here v plays the role of u 1 and u the role of u 2 .…”

Section: Theoremmentioning

confidence: 99%

“…Therefore, it can be interesting when dealing with continuous representations of interval orders too [2,4,5,7,14,15,27] (also dealing with upper or lower semicontinuity [10]), even in order to study the existence (and construction) of a continuous multi-utility representations [8].…”

Section: Introductionmentioning

confidence: 99%

Generalized Debreu’s Open Gap Lemma and Continuous Representability of Biorders

Estevan

2015

Order

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We prove a generalization of Debreu's Open Gap Lemma. Given any subset of the real line, this lemma guarantees the existence of a strictly increasing real function such that all the gaps of the image of the subset are open. Now we extend it to the case of n subsets of the real line and study the existence of a strictly increasing real function such that all the gaps of the image of each set are open. This function does not exist in general so, we characterize the cases in which it exists. This generalization is not equivalent to Debreu's lemma working on the union of the n subsets, neither on the intersection. Then we use it in order to obtain new results on continuous representations of biorders. We also generalize the concept of biorder.

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Continuous Representability of Interval Orders: The Topological Compatibility Setting

Cited by 14 publications

References 40 publications

Searching for a Debreu’s Open Gap Lemma for Semiorders

Searching for a Debreu’s Open Gap Lemma for Semiorders

A Survey on the Mathematical Foundations of Axiomatic Entropy: Representability and Orderings

Generalized Debreu’s Open Gap Lemma and Continuous Representability of Biorders

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