Semicontinuous Planar Total Preorders on Non-Separable Metric Spaces

Arrastia, María Jesús Campión; Candeal, Juan Carlos; Eraso, Esteban Induráin

doi:10.4134/jkms.2009.46.4.701

Cited by 3 publications

(1 citation statement)

References 34 publications

(28 reference statements)

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“…[20][21][22][23] The existence of a pair of upper semicontinuous real-valued functions representing an interval order on a topological space has recently been characterized by Bosi and Zuanon. In that direction, such a characterization, which generalizes previous interesting results presented by Bridges, is the most general one, since the authors do not impose conditions either on the topology or on the two functions that make up the representation.…”

Section: Introductionmentioning

confidence: 99%

Continuous Representability of Interval Orders: The Topological Compatibility Setting

Bosi

Estevan

García

et al. 2015

Int. J. Unc. Fuzz. Knowl. Based Syst.

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In this paper, we go further on the problem of the continuous numerical representability of interval orders defined on topological spaces. A new condition of compatibility between the given topology and the indifference associated to the main trace of an interval order is introduced. Provided that this condition is fulfilled, a semiorder has a continuous interval order representation through a pair of continuous real-valued functions. Other necessary and sufficient conditions for the continuous representability of interval orders are also discussed, and, in particular, a characterization is achieved for the particular case of interval orders defined on a topological space of finite support.

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Section: Introductionmentioning

confidence: 99%

Continuous Representability of Interval Orders: The Topological Compatibility Setting

Bosi

Estevan

García

et al. 2015

Int. J. Unc. Fuzz. Knowl. Based Syst.

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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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Continuous Order Representability Properties of Topological Spaces and Algebraic Structures

Campión¹,

Candeal²,

Induráin³

et al. 2012

Journal of the Korean Mathematical Society

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Semicontinuous Planar Total Preorders on Non-Separable Metric Spaces

Abstract: Abstract. We prove that every non-separable connected metric space can be endowed with a total preorder that is order-isomorphic to a nonrepresentable subset of the lexicographic plane and semicontinuous with respect to the metric topology.

Cited by 3 publications

References 34 publications

Continuous Representability of Interval Orders: The Topological Compatibility Setting

Continuous Representability of Interval Orders: The Topological Compatibility Setting

Searching for a Debreu’s Open Gap Lemma for Semiorders

Continuous Order Representability Properties of Topological Spaces and Algebraic Structures

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