Definition:Upper Closure/Element

Definition
Let $\left({S, \preceq}\right)$ be a ordered set.

Let $a \in S$.

Then we define:


 * ${\bar\uparrow} a := \left\{{b \in S: a \preceq b}\right\}$

That is, ${\bar\uparrow} a$ is the set of all elements of $S$ that succeed $a$.

${\bar\uparrow} a$ is described as the upper closure of $a$ (in $S$).

Also known as
The upper closure of an element is also called the up-set or the principal upper set of that element.

Many texts write ${\uparrow} a$ instead of ${\bar\uparrow} a$, but this clashes with a notation currently common on for strict up-set.

Also see

 * Definition:Lower Closure/Element
 * Definition:Strict Up-Set/Element