Abstract
In this chapter we deal with interval orders and their representation by pairs of real-valued functions. The notion of an interval order is introduced in Section 1, where we also construct a representation when the underlying set is countable. In Section 2 we discuss various topological properties of an interval order on a topological space X. This prepares the way for the case where X is a topological space and we seek representing functions with appropriate continuity properties. Section 3 covers the case where X is a convex subset of N-dimensional euclidean space, when we extend the Arrow-Hahn approach described in Chapter 2. Section 4 contains some representation theorems applicable when X is a topological space. The final section describes Chateauneuf’s theorem, giving necessary and sufficient conditions for the existence of a pair of continuous functions representing an interval order on a topological space.
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© 1995 Springer-Verlag Berlin Heidelberg
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Bridges, D.S., Mehta, G.B. (1995). Interval Orders. In: Representations of Preferences Orderings. Lecture Notes in Economics and Mathematical Systems, vol 422. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-51495-1_6
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DOI: https://doi.org/10.1007/978-3-642-51495-1_6
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-58839-9
Online ISBN: 978-3-642-51495-1
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