Definition:Strict Lower Closure/Element/Also known as

Strict Lower Closure of Element: 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 $\preccurlyeq$ is a well-ordering)
 * 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 $\map s a$ 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 $S$ under a well-ordering.

In particular, see Initial Segment of Natural Numbers.