Abstract
In this paper, we provide a characterization of the existence of a representation of an interval order on a topological space in the general case by means of a pair of continuous functions, when neither the functions nor the topological space are required to satisfy any particular assumptions. Such a characterization is based on a suitable continuity assumption of the binary relation, called weak continuity. In this way, we generalize all the previous results on the continuous representability of interval orders, and also of total preorders, as particular cases.
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Notes
Here, as for interval orders, \(\prec ^*=\prec \circ \precsim \) and \(\prec ^{**}=\precsim \circ \prec \).
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Acknowledgements
Asier Estevan acknowledges financial support from the Ministry of Economy and Competitiveness of Spain under grants MTM2015-63608-P and ECO2015-65031. Gianni Bosi acknowledges financial support from the Istituto Nazionale di Alta Matematica “F. Sever” (Italy).
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Communicated by Juan-Enrique Martinez Legaz.
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Bosi, G., Estevan, A. Continuous Representations of Interval Orders by Means of Two Continuous Functions. J Optim Theory Appl 185, 700–710 (2020). https://doi.org/10.1007/s10957-020-01675-0
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DOI: https://doi.org/10.1007/s10957-020-01675-0