Definition:Weak Initial Segment

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

Let $a \in S$.

Then we define:


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

That is, $S_a$ is the set of all elements of $S$ that precede $a$.

$\bar S_a$ is described as the weak initial segment (of $S$) determined by $a$.

Also known as
Some sources use $\bar s \left({a}\right)$ for $\bar S_a$.

Some sources write ${\downarrow} \left({a}\right)$ for $\bar{S}_a$, and call it the (weak) lower closure of $a$ (in $S$).

However, as the notation leads to confusion with notation currently used on for strict down-set, we write ${\bar\downarrow} \left({a}\right)$ instead.

As the notation varies somewhat from text to text, it is best to define it before using it, whether in a proof or elsewhere.

Also defined as
Some sources use this term only in the context of well-ordered sets.

Also see

 * Initial segment, also called a strict initial segment or strict down-set.
 * Weak Upper Closure