Definition:Strict Upper Closure

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

Let $a \in S$.

Then we define:


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

or alternatively:


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

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

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

Also see

 * Definition:Strict Lower Closure
 * Definition:Weak Upper Closure