Definition:Strict Down-Set

Definition
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 $S$).

Remark
The notation ${\dot\downarrow} T$ is also used to indicate the strict lower set of a subset of $S$.

In most cases there can be no confusion as to which is intended, but if that is ambiguous, the text should clarify.

Also known as
Some call this the 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