Order Isomorphism is Equivalence Relation

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Theorem

Order isomorphism between ordered sets is an equivalence relation.

So any given family of ordered sets can be partitioned into disjoint classes of isomorphic sets.


Proof 1


Follows directly from Identity Mapping is Order Isomorphism.


Follows directly from Inverse of Order Isomorphism is Order Isomorphism.


Follows directly from Composite of Order Isomorphisms is Order Isomorphism.

$\blacksquare$


Proof 2

An ordered set is a relational structure where order isomorphism is a special case of relation isomorphism.

The result follows directly from Relation Isomorphism is Equivalence Relation.

$\blacksquare$


Also see