Definition:Initial Segment

Definition
Let $\left({S, \preceq}\right)$ be a partially ordered set.

Let $a \in S$.

Then we define:


 * $S_a := \left\{{b \in \left({S, \preceq}\right): b \preceq a \land b \ne a}\right\}$

or alternatively:


 * $S_a := \left\{{b \in \left({S, \preceq}\right): b \prec a}\right\}$

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

$S_a$ is described as the initial segment (of $S$) determined by $a$.

Alternative Names
Some sources refer to this concept as a segment.

Some sources reserve this definition exclusively for well-ordered sets, as this is a concept which is usually used in the context of ordinals (which are, by definition, segments of well-ordered sets).

It is also worth noting that the concept of defining the set of all elements which are related to another element crops up throughout the fields of mapping theory and relation theory, defining that set as a segment is usually done only in the context of order theory.

The concept of an initial segment is often (and usually more clearly) referred to by its mundane description: the set of preceding elements.

Notation
Some sources use $s \left({a}\right)$ for $S_a$.

There is no standard convention for this concept. Therefore it is important, before introducing the notation into a thesis, to define it.