Definition:Strict Down-Set

Strict Down-Set of Element
Let $\left({S, \preceq}\right)$ be an ordered set or preordered set.

Let $a \in S$.

Then we define:


 * ${\dot\downarrow} a := \left\{{b \in S: (b \preceq a) \land (b \ne a)}\right\}$

or alternatively:


 * ${\dot\downarrow} a := \left\{{b \in S: b \prec a}\right\}$

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

${\dot\downarrow} a$ is described as the strict down-set of $a$ (in $S$).

Strict Down-Set of Set
Let $\left({S, \preceq}\right)$ be an ordered set or preordered set.

Let $T \subseteq S$.

Then we define:


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

${\dot\downarrow} T$ is described as the strict down-set of $T$ (in $S$).

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.

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 likely to change soon.

Also see

 * Strict Up-Set
 * Lower Closure