Definition:Chain (Order Theory)/Subset Relation

This page is about chain of sets in the context of order theory. For other uses, see chain.

Definition

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 $Y \subseteq X$

Also known as

A chain (of sets) is also known as a nest, particularly in the wider context of class theory.

Some sources use the term tower (of sets).

Examples

Arbitrary Example

Let $A = \set {\set 1, \set {1, 2}, \set {1, 2, 3} }$.

Then $A$ constitutes a chain of sets.

Also see

• Definition:Nest
• Definition:Chain (Order Theory)
• Definition:Nested Sequence

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

Sources

• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): chain: 2.
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): chain: 2.
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): nested sets
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): chain: 2.
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): nested sets
• 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $4$: Superinduction, Well Ordering and Choice: Part $\text I$ -- Superinduction and Well Ordering: $\S 2$ Superinduction and double superinduction