Definition:Upper Closure/Element

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

Let $a \in S$.

The upper closure of $a$ (in $S$) is defined as:


 * $a^\succcurlyeq := \left\{{b \in S: a \preccurlyeq b}\right\}$

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

Also known as
The upper closure of an element $a$ is also known as the up-set of $a$.

The terms weak upper closure and weak up-set are also encountered, so as explicitly to distinguish this from the strict upper closure of $a$.

Also see

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