On finitely generated vector sublattices
Volume 245 / 2019
Abstract
We investigate various questions concerning vector sublattices, $\operatorname {vlt} S$, generated by subsets $S$ of an Archimedean vector lattice $E$. We first prove a distributivity law: $\operatorname {vlt}(X;Y,Z)=\operatorname {vlt}(X,Y)+\nobreakspace {}\operatorname {vlt}(X,Z)$ if $X,Y,Z\subset E$ and $Y\perp Z$, and derive a number of its consequences. We next show that in a topological vector lattice the dimension of the sublattice generated by an analytic set is either $\le \aleph _0$ or $2^{\aleph _0}$, and that the same is true for sublattices generated by at most countable sets in arbitrary vector lattices. In a vector lattice, we characterize those sets that generate $n$-dimensional sublattices and prove that a finite set generates a finite-dimensional sublattice if so does each pair of its elements. We also show that in a uniformly complete vector lattice every principal ideal of infinite dimension contains pairs of positive elements generating $\aleph _0$- as well as $2^{\aleph _0}$-dimensional sublattices. The special case of lattices $C(K)$ is also treated in this respect. Moreover, for a compact set $K\subset \mathbb R^n$ with a nonempty interior, it is shown that the minimal number of functions in $C(K)$ or $C(K)_+$ generating a dense sublattice is $n+1$. We also prove that every (separable) Banach lattice $C(K)$ can be embedded in a discrete (separable) Banach lattice of the same type. Finally, we prove that in a discrete and $\sigma $-Dedekind complete separable $F$-lattice one can always find a pair of positive elements generating a dense sublattice, and we use that result to show that, in general, this is far from being true even in the case of discrete separable $C(K)$ lattices.
