Order Embeddings with Irrational Codomain: Debreu Properties of Real Subsets |
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Authors: | M J Campión J C Candeal E Induráin G B Mehta |
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Institution: | (1) Departamento de Matemáticas, Campus Arrosadía, Universidad Pública de Navarra, 31006 Pamplona, Spain;(2) Facultad de Ciencias Económicas y Empresariales, Departamento de Análisis Económico, Universidad de Zaragoza, c/ Doctor Cerrada 1–3, 50005 Zaragoza, Spain;(3) Departament of Economics, University of Queensland, 4072 Brisbane, Queensland, Australia |
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Abstract: | The objective of this paper is to investigate the role of the set of irrational numbers as the codomain of order-preserving
functions defined on topological totally preordered sets. We will show that although the set of irrational numbers does not
satisfy the Debreu property it is still nonetheless true that any lower (respectively, upper) semicontinuous total preorder
representable by a real-valued strictly isotone function (semicontinuous or not) also admits a representation by means of
a lower (respectively, upper) semicontinuous strictly isotone function that takes values in the set of irrational numbers.
These results are obtained by means of a direct construction. Moreover, they can be related to Cantor’s characterization of the real line to obtain much more general results on the semicontinuous
Debreu properties of a wide family of subsets of the real line.
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Keywords: | total preorders semicontinuous strictly isotone functions irrational numbers Debreu properties |
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