Minimal latticesubspaces
Author:
Ioannis A. Polyrakis
Journal:
Trans. Amer. Math. Soc. 351 (1999), 41834203
MSC (1991):
Primary 46B42, 52A21, 15A48, 53A04
Published electronically:
April 20, 1999
MathSciNet review:
1621706
Fulltext PDF Free Access
Abstract 
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Abstract: In this paper the existence of minimal latticesubspaces of a vector lattice containing a subset of is studied (a latticesubspace of is a subspace of which is a vector lattice in the induced ordering). It is proved that if there exists a Lebesgue linear topology on and is closed (especially if is a Banach lattice with order continuous norm), then minimal latticesubspaces with closed positive cone exist (Theorem 2.5). In the sequel it is supposed that is a finite subset of , where is a compact, Hausdorff topological space, the functions are linearly independent and the existence of finitedimensional minimal latticesubspaces is studied. To this end we define the function where . If is the range of and the convex hull of the closure of , it is proved:  (i)
 There exists an dimensional minimal latticesubspace containing if and only if is a polytope of with vertices (Theorem 3.20).
 (ii)
 The sublattice generated by is an dimensional subspace if and only if the set contains exactly points (Theorem 3.7).
This study defines an algorithm which determines whether a finitedimensional minimal latticesubspace (sublattice) exists and also determines these subspaces.
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Additional Information
Ioannis A. Polyrakis
Affiliation:
Department of Mathematics\ National Technical University of Athens\ Zographou 157 80, Athens, Greece
Email:
ypoly@math.ntua.gr
DOI:
http://dx.doi.org/10.1090/S0002994799023843
PII:
S 00029947(99)023843
Received by editor(s):
March 16, 1997
Published electronically:
April 20, 1999
Additional Notes:
This research was supported by the 1995 PENED program of the Ministry of Industry, Energy and Technology of Greece
Article copyright:
© Copyright 1999
American Mathematical Society
