Definition:Order Type

Definition

Let $\struct {S, \preccurlyeq_1}$ and $\struct {T, \preccurlyeq_2}$ be ordered sets.

Then $S$ and $T$ have the same (order) type if and only if they are order isomorphic.

The order type of an ordered set $\struct {S, \preccurlyeq}$ can be denoted $\map \ot {S, \preccurlyeq}$.

Also defined as

Some sources define an order type on a totally ordered set only.

$\mathsf{Pr} \infty \mathsf{fWiki}$ prefers the more general definition.

Also see

• Results about order types can be found here.

Sources

• 1968: A.N. Kolmogorov and S.V. Fomin: Introductory Real Analysis ... (previous) ... (next): $\S 3.3$: Ordered sets. Order types
• 1996: Winfried Just and Martin Weese: Discovering Modern Set Theory. I: The Basics ... (previous) ... (next): Part $1$: Not Entirely Naive Set Theory: Chapter $2$: Partial Order Relations
• 2000: James R. Munkres: Topology (2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 3$: Relations