Definition:Lower Closure/Element

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

Let $a \in S$.

The lower closure of $a$ (in $S$) is defined as:


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

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

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 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 $\left({S, \preccurlyeq}\right)$ is a well-ordered set, 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 on a general ordered set.

Also see

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