Order Isomorphism between Posets an Equivalence
Contents |
Theorem
Order isomorphism between posets is an equivalence.
So any given family of posets can be partitioned into disjoint classes of isomorphic sets.
So, two isomorphic posets can be regarded as identical where it is the structure of the partial ordering that is important rather than the elements themselves.
Two isomorphic posets obviously have the same power, as there is a bijection between them by definition.
Proof 1
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Follows directly from the fact that the identity mapping is an order isomorphism.
Follows directly from the fact that the inverse of an order isomorphism is itself an order isomorphism.
Follows directly from the fact that the composite of two order isomorphisms is itself an order isomorphism.
$\blacksquare$
Proof 2
A poset is a relational structure where order isomorphism is a special case of relation isomorphism.
The result follows directly from Relation Isomorphism is an Equivalence.
Sources
- A.N. Kolmogorov and S.V. Fomin‎: Introductory Real Analysis (1968): $\S 3.2$
