Definition:Lower Closure/Element/Also known as

Lower Closure of Element: Also known as
The lower closure of an element $a$ is also known as:
 * the down-set of $a$
 * the down set of $a$
 * the lower set of $a$
 * the lower section of $a$
 * the set of preceding elements to $a$

The terms weak lower closure and weak down-set are also encountered, so as explicitly to distinguish this from the strict lower closure of $a$.

When $\preccurlyeq$ is a well-ordering, the term weak initial segment is often used, and defined as a separate concept in its own right.

The notations $S_a$ or $\bar S_a$ are frequently then seen.

Some authors use the term (weak) initial segment to refer to the lower closure a general ordering.