Bands in partially ordered vector spaces with order unit

Bands in partially ordered vector spaces with order unit In an Archimedean directed partially ordered vector space X, one can define the concept of a band in terms of disjointness. Bands can be studied by using a vector lattice cover Y of X. If X has an order unit, Y can be represented as a subspace of $$C(\Omega )$$ C ( Ω ) , where $$\Omega $$ Ω is a compact Hausdorff space. We characterize bands in X, and their disjoint complements, in terms of subsets of $$\Omega $$ Ω . We also analyze two methods to extend bands in X to $$C(\Omega )$$ C ( Ω ) and show how the carriers of a band and its extensions are related. We use the results to show that in each n-dimensional partially ordered vector space with a closed generating cone, the number of bands is bounded by $$\frac{1}{4}2^{2^n}$$ 1 4 2 2 n for $$n\ge 2$$ n ≥ 2 . We also construct examples of $$(n+1)$$ ( n + 1 ) -dimensional partially ordered vector spaces with $$\left( {\begin{array}{c}2n\\ n\end{array}}\right) +2$$ 2 n n + 2 bands. This shows that there are n-dimensional partially ordered vector spaces that have more bands than an n-dimensional Archimedean vector lattice when $$n\ge 4$$ n ≥ 4 . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

Bands in partially ordered vector spaces with order unit

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Publisher
Springer Journals
Copyright
Copyright © 2015 by The Author(s)
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-014-0311-7
Publisher site
See Article on Publisher Site

Abstract

In an Archimedean directed partially ordered vector space X, one can define the concept of a band in terms of disjointness. Bands can be studied by using a vector lattice cover Y of X. If X has an order unit, Y can be represented as a subspace of $$C(\Omega )$$ C ( Ω ) , where $$\Omega $$ Ω is a compact Hausdorff space. We characterize bands in X, and their disjoint complements, in terms of subsets of $$\Omega $$ Ω . We also analyze two methods to extend bands in X to $$C(\Omega )$$ C ( Ω ) and show how the carriers of a band and its extensions are related. We use the results to show that in each n-dimensional partially ordered vector space with a closed generating cone, the number of bands is bounded by $$\frac{1}{4}2^{2^n}$$ 1 4 2 2 n for $$n\ge 2$$ n ≥ 2 . We also construct examples of $$(n+1)$$ ( n + 1 ) -dimensional partially ordered vector spaces with $$\left( {\begin{array}{c}2n\\ n\end{array}}\right) +2$$ 2 n n + 2 bands. This shows that there are n-dimensional partially ordered vector spaces that have more bands than an n-dimensional Archimedean vector lattice when $$n\ge 4$$ n ≥ 4 .

Journal

PositivitySpringer Journals

Published: Jul 9, 2015

References

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