Definition:Lower Closure/Set

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

Let $T \subseteq S$.

Then we define:


 * ${\downarrow} T := \bigcup \left\{{ {\bar\downarrow} t: t \in T }\right\}$

where ${\bar\downarrow} t$ is the lower closure of $t$.

That is:
 * ${\downarrow} T:= \{ l \in S: \exists t \in T: l \preceq t \}$

${\downarrow} T$ is described as the lower closure of $T$ (in $S$).

Remark
The notation ${\downarrow} a$ may also be used to refer to the lower closure of an element of an ordered set.

This usage, however, clashes with a notation currently common on for the strict down-set of an element, so the lower closure of an element $a$ should be written ${\bar\downarrow} a$.

Also see

 * Upper Closure:Set
 * Strict Down-Set: Set