Order Isomorphism is Equivalence Relation

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.

$\blacksquare$