Definition:Lower Closure/Element

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

Let $a \in S$.

Then we define:


 * ${\bar\downarrow} a := \left\{{b \in S: b \preceq a}\right\}$

That is, ${\bar\downarrow} a$ is the set of all elements of $S$ that precede $a$.

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

Also known as
The lower closure of an element is also called the down-set or the principle lower set of that element.

It is sometimes also called the weak initial segment of the element, but that term and its accompanying notation are usually reserved for well-ordered sets.

Many texts write ${\downarrow} a$ instead of ${\bar\downarrow} a$, but this clashes with a notation currently common on for strict down-set.

Also see

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