Exercise 12, § 1

Let E be a fully lattice-ordered space. Show that there exists a family $(u_\iota)$ of elements $> 0$ of E such that $u_\iota$ and $u_\kappa$ are alien for every pair of distinct indices $\iota, \kappa$, and such that for every $x > 0$ in E there exists at least one index $\iota$ such that $\inf(x, u_\iota) > 0$ (use Zorn’s lemma). From this, deduce that for every $x \geq 0$ there exists one and only one family $(x_\iota)$ of elements $\geq 0$ of E such that, for every $\iota$, $x_\iota$ belongs to the band $B_\iota$ generated by $u_\iota$, and such that $x = \sum_\iota x_\iota$ for the order structure of E (take $x_\iota$ to be the component of $x$ in the band $B_\iota$). Conversely, every family $(x_\iota)$ of elements $\geqslant 0$ of E that is bounded above, such that $x_\iota \in B_\iota$ for all $\iota$, is summable in E (Exer. 11).