Semiorders with separability properties

Candeal, Juan Carlos; Estevan, Asier; García, Javier Gutiérrez; Induráin, Esteban

doi:10.1016/j.jmp.2013.01.003

Cited by 22 publications

(16 citation statements)

References 52 publications

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“…It is the case, for instance, of semiorder structures [13,20,22,23,25,26,29]. In this case, the order structure is represented by means of a single function u (also named utility function) and a strictly positive constant k, such that x ≺ y if and only if u(x) + k < u(y) [29].…”

Section: Discussionmentioning

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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Section: Discussionmentioning

confidence: 99%

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

Estevan

2015

Order

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“…Indeed, let a ∈ G + * . Write a in normal form as in (6). By definition, m s > 0 from which it follows that g(α 1 ) ≺ a (this is obvious if s = 1; if s = 1 observe that −g(α 1 ) + a = (m 1 − 1) g(α 1 ) + m 2 g(α 2 ) + · · · + m s g(α s ) is under normal form with m s > 0 hence is positive).…”

Section: Clifford's Example and A Proof Of Theorem 13mentioning

confidence: 99%

“…This paper is about the interactions between order and group operation in an ordered group. It focuses on semiorders and interval orders, two notions whose role in Mathematical Psychology is well documented through books and papers, see for example [1,4,5,31] and [2,3,6,7,10,11,15,25,26,28,34,35,40]. Interval orders are much more general than semiorders.…”

Section: Introductionmentioning

confidence: 99%

Interval orders, semiorders and ordered groups

Pouzet

Zaguia

2019

Journal of Mathematical Psychology

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We prove that the order of an ordered group is an interval order if and only if it is a semiorder. Next, we prove that every semiorder is isomorphic to a collection J of intervals of some totally ordered abelian group, these intervals being of the form [x, x + α[ for some positive α. We describe ordered groups such that the ordering is a semiorder and we introduce threshold groups generalizing totally ordered groups. We show that the free group on finitely many generators and the Thompson group F can be equipped with a compatible semiorder which is not a weak order. On another hand, a group introduced by Clifford cannot.

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“…The Scott-Suppes representability of semiorders is finally characterized as follows (see, e.g., [117][118][119][120]…”

Section: Remark 23mentioning

confidence: 99%

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

Campión

Gómez‐Polo

Induráin

et al. 2018

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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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Semiorders with separability properties

Cited by 22 publications

References 52 publications

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

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

Interval orders, semiorders and ordered groups

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

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