Corollary

— Let $a$ be an element of a fully lattice-ordered space $E$, $B_a$ the band generated by $a$, $B'_a$ the band of elements alien to $a$. For every element $x \geqslant 0$ of $E$, the component of $x$ in $B_a$ (for the decomposition of $E$ as the ordered direct sum of $B_a$ and $B'_a$) is equal to $\sup_{n \in \mathbf{N}} (\inf(n|a|, x))$.

This follows from Proposition 6, applied to $M = \{a\}$, and Proposition 5.

Note that the bands generated by $a$ and $|a|$ are identical. If $a$ and $b$ are two elements of $E$ that are alien to each other, and if $A$ and $B$ are the bands generated by $a$ and $b$, respectively, then every element of $A$ is alien to every element of $B$; for, $b$ belongs to the band $A'$ of elements alien to $a$, whence $B \subset A'$, and, by Theorem 1, every element of $A$ is alien to every element of $A'$.