Definition:Order Isomorphism/Isomorphic Sets
< Definition:Order Isomorphism(Redirected from Definition:Isomorphic Ordered Sets)
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Definition
Two ordered sets $\struct {S, \preceq_1}$ and $\struct {T, \preceq_2}$ are (order) isomorphic if and only if there exists such an order isomorphism between them.
Hence $\struct {S, \preceq_1}$ is described as (order) isomorphic to (or with) $\struct {T, \preceq_2}$, and vice versa.
This may be written $\struct {S, \preceq_1} \cong \struct {T, \preceq_2}$.
Where no confusion is likely to arise, it can be abbreviated to $S \cong T$.
Also see
- Results about order isomorphisms can be found here.
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 14$: Orderings
- 1968: A.N. Kolmogorov and S.V. Fomin: Introductory Real Analysis ... (previous) ... (next): $\S 3.2$: Order-preserving mappings. Isomorphisms
- 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 7$
- 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: Definition $5$