Definition:Lower Closure/Set

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

Let $T \subseteq S$.

The lower closure of $T$ (in $S$) is defined as:


 * $T^\preccurlyeq := \bigcup \left\{{t^\preccurlyeq: t \in T}\right\}$

where $t^\preccurlyeq$ is the lower closure of $t$.

That is:
 * $T^\preccurlyeq := \left\{{l \in S: \exists t \in T: l \preccurlyeq t}\right\}$

Also see

 * Definition:Lower Closure of Element


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


 * Lower Closure is Closure Operator
 * Lower Closure is Smallest Containing Lower Set