Definition:Lower Closure/Element

Definition
Let $\struct {S, \preccurlyeq}$ be an ordered set.

Let $a \in S$.

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


 * $a^\preccurlyeq := \set {b \in S: b \preccurlyeq a}$

That is, $a^\preccurlyeq$ is the set of all elements of $S$ that precede $a$.

Also see

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