Definition:Weak Initial Segment

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

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

There is no standard convention for this concept. Therefore it is important, before introducing the notation into a thesis, to define it.

Also see

 * Initial segment, also called a strict initial segment.