Definition:Strict Down-Set

Remark
In most cases, there can be no confusion as to whether the strict down-set of an element or of a set is intended. Where that is ambiguous, it should be indicated in the text.

Also known as
Some call this a strict down set or a strict lower set.

Some place the arrow after the element instead of before.

Some use $D(a)$ instead of ${\dot\downarrow} a$.

On, this is often called strict lower closure and written ${\downarrow}a$, but this is changing.

Also see

 * Strict Up-Set
 * Lower Closure