\begin{definition}[Definition:Initial Segment]
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 $S$)'''.
\end{definition}