Definition:Chain (Set Theory)

Definition

Let $\left({S, \preceq}\right)$ 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.

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.

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): Entry: chain: 1.
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Entry: chain: 1.