Definition:Initial Segment

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

Let $a \in S$.

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


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

which can also be rendered as:


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

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

Some sources refer to this concept as a segment.

Some sources refer to this concept as a section.

When it is necessary to distinguish between this and a weak initial segment, this is called a strict initial segment.

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

In the context of a general ordered set, the concept is used more broadly, and there are far too many synonyms for this and related concepts.

On in particular, it is referred to as (strict) lower closure.

In such a context, the notation $a^\prec$ is used on to denote the initial segment.

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

Also defined as
Some sources use the term initial segment to refer to sets with a certain property relative to a relation. Terminology for such sets is currently not fixed on.

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. However, defining that set as a segment is usually done only in the context of order theory.

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