Minimal lattice-subspaces

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.