Definition:Upper Closure/Element

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

Let $a \in S$.

Then we define:


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

That is, $\mathop{\bar \uparrow} \left({a}\right)$ is the set of all elements of $S$ that succeed $a$.

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

Also see

 * Weak Lower Closure
 * Strict Upper Closure