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An order-theoretic approach to interval analysis. (English) Zbl 0829.65053

Summary: Interval analysis makes heavy use of inclusion order for intervals, and relies on the notion of monotonicity of functions for this order. It turns out that \(({\mathbf I}{\mathbf R}, \supseteq)\), where \({\mathbf I} {\mathbf R}\) is the set of Moore intervals plus \([-\infty, +\infty]\), is a complete partial order (cpo). Then it is appropriate to use the theory of cpo’s in order to fully formalize the role of inclusion order.
This work develops basic concepts of interval analysis within this order- theoretic framework. There is a natural topology for cpo’s, the Scott topology. It is shown that the Scott topology on \({\mathbf I} {\mathbf R}\) is compatible with both inclusion-monotonicity and the usual topology on the real line.

MSC:

65G30 Interval and finite arithmetic
06F30 Ordered topological structures
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