Definition:Initial Segment

Definition
Let $\struct {S, \preceq}$ be a well-ordered set.

Let $a \in S$.

The initial segment (of $S$) determined by $a$ is defined as:


 * $S_a := \set {b \in S: b \preceq a \land b \ne a}$

which can also be rendered as:


 * $S_a := \set {b \in S: b \prec a}$

That is, $S_a$ is the set of all elements of $S$ that strictly precede $a$.

That is, $S_a$ is the strict lower closure of $a$ (in $S$).

By extension, $S_a$ is described as an initial segment (of $S$).

Also see

 * Definition:Initial Segment of Natural Numbers


 * Definition:Weak Initial Segment


 * Definition:Strict Lower Closure
 * Definition:Strict Upper Closure


 * Definition:Lower Closure
 * Definition:Upper Closure