2004
DOI: 10.1016/j.jmateco.2003.06.002
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The Debreu Gap Lemma and some generalizations

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Cited by 24 publications
(29 citation statements)
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“…In Herden and Mehta [12] the Debreu Open Gap Lemma has been discussed extensively. In particular, its value in any theory of continuous order-preserving functions the codomain of which is not necessarily the real line has been underlined.…”
Section: Debreu Setsmentioning
confidence: 99%
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“…In Herden and Mehta [12] the Debreu Open Gap Lemma has been discussed extensively. In particular, its value in any theory of continuous order-preserving functions the codomain of which is not necessarily the real line has been underlined.…”
Section: Debreu Setsmentioning
confidence: 99%
“…Then a fundamental problem in mathematical utility theory is the problem of determining necessary and sufficient conditions that guarantee the existence of some real-valued orderpreserving (strictly increasing) function f on (X, ) that also preserves the mathematical structure on (X, ). In Herden and Mehta [12] it has been underlined by many examples that in a more applicable approach to mathematical utility theory the real line often is not the appropriate codomain of f . This means that the real line in many concrete cases should be replaced by a codomain that is more carefully adapted to the particular considered representation problem.…”
Section: Introductionmentioning
confidence: 99%
“…Also, from a purely mathematical point of view there is no reason or justification for studying only the very special case where R is the codomain. For further discussion of these ideas we refer the reader to [4] and [14]. Therefore, it is desirable to develop a general theory of representations of ordered structures in which the codomain is not necessarily the set of real numbers.…”
Section: Introductionmentioning
confidence: 99%
“…One way to proceed is to consider codomains such as the long line or the lexicographic plane as in [14]. In this paper we follow a somewhat different approach by studying codomains of strictly isotone functions that are subsets of R but not necessarily order-isomorphic to R. (For initial results in this direction see [8] where order-monomorphisms are studied with codomains consisting of the rational numbers and the real algebraic numbers.…”
Section: Introductionmentioning
confidence: 99%
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