Abstract
In 1956 R. D. Luce introduced the notion of a semiorder to deal with indifference relations in the representation of a preference. During several years the problem of finding a utility function was studied until a representability characterization was found. However, there was almost no results on the continuity of the representation. A similar result to Debreu’s Lemma, but for semiorders was never achieved. In the present paper we propose a characterization for the existence of a continuous representation (in the sense of Scott-Suppes) for bounded semiorders. As a matter of fact, the weaker but more manageable concept of \(\varepsilon \)-continuity is properly introduced for semiorders. As a consequence of this study, a version of the Debreu’s Open Gap Lemma is presented (but now for the case of semiorders) just as a conjecture, which would allow to remove the open-closed and closed-open gaps of a subset \(S\subseteq \mathbb {R}\), but now keeping the constant threshold, so that \(x+1<y\) if and only if \(g(x)+1<g(y) \, (x,y\in S)\).
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Notes
- 1.
Observe that \(\tau \) does not coincide with the induced topology \( \tau _{u|_X}\) inherited from the usual topology \(\tau _u\) on \(\mathbb {R}\). As a matter of fact \(\tau \subsetneq \tau _{u|_X}\).
- 2.
J denotes here a directed set of indices. Since this does not lead to confusion, we will use the same notation ‘<’ of the order on the real numbers than for the partial order on the set of indices J.
- 3.
Here, notice that the sequence \(\{u_n\}_{n\in \mathbb {N}}\) is a Cauchy sequence with respect to the sup norm.
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Acknowledgements
The author acknowledges financial support from the Ministry of Economy and Competitiveness of Spain under grants MTM2015-63608-P and ECO2015-65031.
I am grateful for the helpful advice of the anonymous referees, to whom I am indebted for the detailed reading of previous versions of the manuscript and their very helpful suggestions and comments.
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Estevan, A. (2020). Searching for a Debreu’s Open Gap Lemma for Semiorders. In: Bosi, G., Campión, M., Candeal, J., Indurain, E. (eds) Mathematical Topics on Representations of Ordered Structures and Utility Theory. Studies in Systems, Decision and Control, vol 263. Springer, Cham. https://doi.org/10.1007/978-3-030-34226-5_5
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