# Definition:Chain (Order Theory)

*This page is about Chain in the context of Order Theory. For other uses, see Chain.*

## Definition

Let $\struct {S, \preceq}$ be an ordered set.

A **chain in $S$** is a totally ordered subset of $S$.

Thus a totally ordered set is itself a **chain** in its own right.

### Chain of Sets

An important special case of a chain is where the ordering in question is the subset relation:

Let $S$ be a set.

Let $\powerset S$ be its power set.

Let $N \subseteq \powerset S$ be a subset of $\powerset S$.

Then $N$ is a **chain (of sets)** if and only if:

- $\forall X, Y \in N: X \subseteq Y$ or $Y \subseteq X$

## Length

Let $T$ be a chain in $S$.

Let $T$ be finite and non-empty.

The **length** of the chain $T$ is its cardinality minus $1$.

## Also defined as

Some sources use the term **chain** to mean the same thing as totally ordered set.

While this is perfectly valid, as there is no source of confusion here, such usage is surprisingly uncommon.

## Also see

- Results about
**chains**in the context of**order theory**can be found**here**.

## Sources

- 1955: John L. Kelley:
*General Topology*... (previous) ... (next): Chapter $0$: Orderings - 1960: Paul R. Halmos:
*Naive Set Theory*... (previous) ... (next): $\S 16$: Zorn's Lemma - 1964: Steven A. Gaal:
*Point Set Topology*... (previous) ... (next): Introduction to Set Theory: $3$. The Axiom of Choice and Its Equivalents - 1975: T.S. Blyth:
*Set Theory and Abstract Algebra*... (previous) ... (next): $\S 7$ - 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next):**chain**:**1.**