Definition:Strict Lower Closure/Element

Definition
Let $\left({S, \preccurlyeq}\right)$ be a 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, $a^\succ$ is the set of all elements of $S$ that strictly precede $a$.

Also known as
The strict lower closure of an element $a$ is also known as the strict down-set of $a$.

When $\left({S, \preccurlyeq}\right)$ is a well-ordered set, the term initial segment is used, and defined as a separate concept in its own right.

Also see

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