Proposition 2

— Let $E$ be a Riesz space equipped with a Hausdorff topology that is compatible with its ordered vector space structure (TVS, II, §2, No. 7). If, for every set $H \subset E$ that is bounded above and directed for the relation $\leqslant$, the section filter of $H$ is convergent, then $E$ is fully lattice-ordered.

Indeed, one knows that the limit of the section filter of $H$ is the supremum of $H$ in $E$ (TVS, II, §2, No. 7, Prop. 18).