Abstract
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.
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Acknowledgments
O. van Gaans acknowledges the support by a ‘VIDI subsidie’ (639.032.510) of the Netherlands Organisation for Scientific Research (NWO).
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Kalauch, A., Lemmens, B. & van Gaans, O. Riesz completions, functional representations, and anti-lattices. Positivity 18, 201–218 (2014). https://doi.org/10.1007/s11117-013-0240-x
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DOI: https://doi.org/10.1007/s11117-013-0240-x
Keywords
- Anti-lattice
- Disjointness
- Embedding
- Functional representation
- Partially ordered vector space
- Riesz completion
- Riesz homomorphism