Definition:Strict Upper Closure/Element

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

Let $a \in S$.

The strict upper closure of $a$ (in $S$) is defined as:
 * $a^\succ := \left\{{b \in S: a \preccurlyeq b \land a \ne b}\right\}$

or:
 * $a^\succ := \left\{{b \in S: a \prec b}\right\}$

That is, $a^\succ$ is the set of all elements of $S$ that strictly succeed $a$.

Also see

 * Definition:Strict Upper Closure of Subset
 * Definition:Upper Closure of Element
 * Definition:Strict Lower Closure of Element