Minimal lattice-subspaces

Author:
Ioannis A. Polyrakis

Journal:
Trans. Amer. Math. Soc. **351** (1999), 4183-4203

MSC (1991):
Primary 46B42, 52A21, 15A48, 53A04

DOI:
https://doi.org/10.1090/S0002-9947-99-02384-3

Published electronically:
April 20, 1999

MathSciNet review:
1621706

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Abstract: In this paper the existence of minimal lattice-subspaces of a vector lattice containing a subset of is studied (a lattice-subspace 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 lattice-subspaces 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 finite-dimensional minimal lattice-subspaces 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 lattice-subspace 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).

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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:
https://doi.org/10.1090/S0002-9947-99-02384-3

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