Riesz completions, functional representations, and anti-lattices

Riesz completions, functional representations, and anti-lattices We show that the Riesz completion of an Archimedean partially ordered vector space $$X$$ with unit can be represented as a norm dense Riesz subspace of the smallest functional representation of $$X.$$ This yields a convenient way to find the Riesz completion. To illustrate the method, the Riesz completions of spaces ordered by Lorentz cones, cones of symmetric positive semi-definite matrices, and polyhedral cones are determined. We use the representation to analyse the existence of non-trivial disjoint elements and link the absence of such elements to the notion of anti-lattice. One of the results is a geometric condition on the dual cone of a finite dimensional partially ordered vector space $$X$$ that ensures that $$X$$ is an anti-lattice. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

Riesz completions, functional representations, and anti-lattices

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Springer Basel
Copyright © 2013 by Springer Basel
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
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