Abstract
A notion of disjointness in arbitrary partially ordered vector spaces is introduced by calling two elements x and y disjoint if the set of all upper bounds of x + y and −x − y equals the set of all upper bounds of x − y and −x + y. Several elementary properties are easily observed. The question whether the disjoint complement of a subset is a linear subspace appears to be more difficult. It is shown that in directed Archimedean spaces disjoint complements are always subspaces. The proof relies on theory on order dense embedding in vector lattices. In a non-Archimedean directed space even the disjoint complement of a singleton may fail to be a subspace. According notions of disjointness preserving operator, band, and band preserving operator are defined and some of their basic properties are studied.
Similar content being viewed by others
References
Abramovich, Y.A. and Kitover, A.K.: Inverses of disjointness preserving operators, Mem. Am. Math. Soc. 143 (679), 2000.
Aliprantis, C.D. and Burkinshaw, O.: Locally solid Riesz spaces, Academic Press, New York, San Francisco, London, 1978.
Aliprantis, C.D. and Burkinshaw, O.: Positive Operators, Academic Press Inc., London, 1985.
Buskes, G. and Van Rooij, A.C.M.: The vector lattice cover of certain partially ordered groups, J. Austral. Math. Soc. (Series A) 54 (1993), 352–367.
Van Gaans, O.: Seminorms on ordered vector spaces that extend to Riesz seminorms on larger Riesz spaces, Indag. Mathem., N.S. 14(1) (2003), 15–30.
Van Haandel, M.: Completions in Riesz Space Theory, Ph.D. thesis, University of Nijmegen, 1993.
Luxemburg, W.A.J. and Zaanen, A.C.: Riesz spaces I, North-Holland Publishing Company, Amsterdam, 1971.
Meyer-Nieberg, P.: Banach lattices, Springer Verlag, Berlin, Heidelberg, 1991.
Vulikh, B.Z.: Introduction to the Theory of Cones in Normed Spaces (Russian), Izdat. Kalinin University, Kalinin, 1977.
Vulikh, B.Z.: Special Topics in the Geometry of Cones in Normed Spaces (Russian), Izdat. Kalinin University, Kalinin, 1978.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Gaans, O., Kalauch, A. Disjointness in Partially Ordered Vector Spaces. Positivity 10, 573–589 (2006). https://doi.org/10.1007/s11117-005-0015-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11117-005-0015-0