Definition 4

In a fully lattice-ordered space $E$, a linear subspace $B$ of $E$ is said to be a band if it satisfies the following conditions: 1) the relations $x \in B, y \in E$ and $|y| \leq |x|$ imply $y \in B$; 2) for every nonempty subset $X$ of $B$ that is bounded above in $E$, the supremum $\sup X$ of $X$ in $E$ belongs to $B$.