Definition 2

In an ordered set E, a subset of E such that the relations $x \in \mathrm{S}$, $y \in \mathrm{E}$, and $y \leqslant x$ imply $y \in \mathrm{S}$ is called a segment of E.

Clearly, every intersection and union of segments of E is a segment of E. If S is a segment of E, every segment of S is also a segment of E. The set E itself and the empty set are segments of E.