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I am trying to find ways to prove the Archimedian property of a certain vector lattice and got stuck on the following type of problem.

I feel the statement below (or in fact weaker versions) should be provable, but I am not being very successful so far. I am posting this question here to get suggestions and/or counterexamples, thanks!

Statement: Let $A$ be a vector lattice (wikipedia entry) and let $B\subseteq A$ a subset of $A$ which:

1) B is closed under vector spaces operations (of $A$), i.e.: (i) if $b_1,b_2\in B$ then $b_1 + b_2 \in B$, and (ii) if $b\in B$ then $r b\in B$ for all $r\in\mathbb{R}$. In other words, $B$ is a partially orderd vector subspace of $A$, but $B$ is not necessarily a lattice.

2) $B$ generates $A$, i.e., every element $a\in A$ is expressible as a finite combination of meet and joins from $B$: $a = \bigvee_i \bigwedge_j b_{i,j}$.

3) $B$ is Archimedean, in the sense that it does not have infinitesimals (except $0$): for all $b,b^\prime\geq0$ in $B$, if for all $n\in\mathbb{N}$ it holds that $nb \leq b^\prime$, then $b=0$.

Under these assumptions it follows that $A$ is Archimedean as a vector lattice. I.e., for all $a,a^\prime\geq0$ in $A$, if for all $n$, $n a \leq a^\prime$ then $a=0$.

end of Statement

As I said, I have not been able to prove this so far. Perhaps there is some counterexample?

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  • $\begingroup$ Do you have an example motivating this question? $\endgroup$ – user458276 Mar 16 at 22:41
  • $\begingroup$ Any examples would need to be infinite dimensional (see math.leidenuniv.nl/scripties/MasterJongen.pdf). Also, since you’d have to a finite combination of meets and joins of elements of $B$ to produce any elements of $A$, it seems unlikely that there are any finite proper subsets $B$ that satisfy (2). $\endgroup$ – user458276 Mar 17 at 0:41

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