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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Minimal lattice-subspaces


Author: Ioannis A. Polyrakis
Journal: Trans. Amer. Math. Soc. 351 (1999), 4183-4203
MSC (1991): Primary 46B42, 52A21, 15A48, 53A04
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 $E$ containing a subset $B$ of $E_+$ is studied (a lattice-subspace of $E$ is a subspace of $E$ which is a vector lattice in the induced ordering). It is proved that if there exists a Lebesgue linear topology $\tau$ on $E$ and $E_+$ is $\tau$-closed (especially if $E$ is a Banach lattice with order continuous norm), then minimal lattice-subspaces with $\tau$-closed positive cone exist (Theorem 2.5).

In the sequel it is supposed that $B=\{x_1,x_2,\ldots,x_n\}$ is a finite subset of $C_+(\Omega)$, where $\Omega$ is a compact, Hausdorff topological space, the functions $x_i$ are linearly independent and the existence of finite-dimensional minimal lattice-subspaces is studied. To this end we define the function $\beta(t) = \frac{r(t)}{\|r(t)\|_1}$ where $r(t) = \big(x_1(t),x_2(t),\ldots,x_n(t)\big)$. If $R(\beta)$ is the range of $\beta$ and $K$ the convex hull of the closure of $R(\beta)$, it is proved:

(i)
There exists an $m$-dimensional minimal lattice-subspace containing $B$ if and only if $K$ is a polytope of $\mathbb{R}^n$ with $m$ vertices (Theorem 3.20).
(ii)
The sublattice generated by $B$ is an $m$-dimensional subspace if and only if the set $R(\beta)$ contains exactly $m$ points (Theorem 3.7).
This study defines an algorithm which determines whether a finite-dimensional minimal lattice-subspace (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/S0002-9947-99-02384-3
PII: S 0002-9947(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