# 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 $S$)**.

### Class Theoretical Definition

In the context of class theory, the definition follows the same lines:

Let $A$ be a class.

Let $\preceq$ be a well-ordering on $A$.

Let $a \in A$.

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

- $A_a := \set {b \in S: b \preceq a \land b \ne a}$

which can also be rendered as:

- $A_a := \set {b \in S: b \prec a}$

## 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**.

Some sources use the term **strict lower closure**, but $\mathsf{Pr} \infty \mathsf{fWiki}$ prefers to reserve that term for when the ordering in question is not necessarily a well-ordering.

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 $\mathsf{Pr} \infty \mathsf{fWiki}$ in particular, it is referred to as (strict) lower closure.

In such a context, the notation $a^\prec$ is used on $\mathsf{Pr} \infty \mathsf{fWiki}$ to denote the **initial segment**.

Some sources use $\map s a$ for $S_a$.

## Also defined as

Some sources use the term **initial segment** to refer to sets with a certain property with respect to a relation.

Terminology for such sets is currently not fixed on $\mathsf{Pr} \infty \mathsf{fWiki}$.

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

- Results about
**initial segments**can be found**here**.

## Sources

- 1993: Keith Devlin:
*The Joy of Sets: Fundamentals of Contemporary Set Theory*(2nd ed.) ... (previous) ... (next): $\S 1$: Naive Set Theory: $\S 1.7$: Well-Orderings and Ordinals - 2000: James R. Munkres:
*Topology*(2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 4$: The Integers and the Real Numbers