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.
KeywordsTopological Space Lower Semicontinuous Continuous Representation Interval Order Strict Partial Order
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