Definition:Strict Up-Set

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

Let $a \in S$.

Then we define:


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

or alternatively:


 * ${\dot\uparrow} a := \left\{{b \in S: a \prec b}\right\}$

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

${\dot\uparrow} a$ is described as the strict up-set of $a$ (in $S$).

Remark
The notation ${\dot\uparrow} T$ is also used to denote the strict upper set of a subset of $S$.

In most cases there can be no confusion as to which is intended, but when there can be, the specific meaning should be indicated in the text.

Also known as
Some call this a strict up set.

Some place the arrow after the element instead of before.

Some use $U(a)$ instead of ${\dot\uparrow} a$.

On, this is often called strict upper closure and written ${\uparrow}a$, but this is likely to change soon.

Also see

 * Strict Down-Set
 * Upper Closure