Definition:Strict Lower Closure/Element

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

Let $a \in S$.

The strict lower closure of $a$ (in $S$) is defined as:
 * $a^\prec := \left\{{b \in S: b \preccurlyeq a \land a \ne b}\right\}$

or:
 * $a^\prec := \left\{{b \in S: b \prec a}\right\}$

That is, $a^\prec$ is the set of all elements of $S$ that strictly precede $a$.

Also known as
The strict lower closure of an element $a$ also goes by the names:
 * strict down-set
 * strict down set
 * initial segment (particularly when $\left({S, \preccurlyeq}\right)$ is a well-ordered set)
 * strict initial segment
 * set of (strictly) preceding elements to $a$

The term (strict) initial segment is usually seen in discussion of the properties of ordinals.

In this context, the notation $S_a$ or $s \left({a}\right)$ can often be found for $a \in S$.

On, the term an initial segment of $S$ is specifically reserved for the strict lower closure of some element $a$ of $S$.

In particular, see Initial Segment of Natural Numbers

Also see

 * Definition:Lower Closure of Element
 * Definition:Strict Lower Closure of Subset
 * Definition:Strict Upper Closure of Element