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 $a$.

Also known as

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

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

However, as the notation leads to confusion with strict lower closure, we prefer to write $\mathop{\bar \downarrow} \left({a}\right)$ instead, mimicking the distinction present in the other notations.

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 or strict lower closure.
• Weak Upper Closure

Sources

• Paul R. Halmos: Naive Set Theory (1960)... (previous)... (next): $\S 14$: Order
• Keith Devlin: The Joy of Sets: Fundamentals of Contemporary Set Theory (1993): $\S 1.7$