Duals of Isomorphic Ordered Sets are Isomorphic

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Theorem

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

Let $\struct {S, \succcurlyeq_1}$ and $\struct {T, \succcurlyeq_2}$ be the dual ordered sets of $\struct {S, \preccurlyeq_1}$ and $\struct {T, \preccurlyeq_2}$ respectively.


Let $f: \struct {S, \preccurlyeq_1} \to \struct {T, \preccurlyeq_2}$ be an order isomorphism.

Then $f: \struct {S, \succcurlyeq_1} \to {T, \succcurlyeq_2} $ is also an order isomorphism.


Proof

\(\displaystyle \forall x, y \in S: \ \ \) \(\displaystyle x\) \(\succcurlyeq_1\) \(\displaystyle y\)
\(\displaystyle \leadstoandfrom \ \ \) \(\displaystyle y\) \(\preccurlyeq_1\) \(\displaystyle x\) Definition of Dual Ordering
\(\displaystyle \leadstoandfrom \ \ \) \(\displaystyle \map f y\) \(\preccurlyeq_2\) \(\displaystyle \map f x\) Definition of Order Isomorphism
\(\displaystyle \leadstoandfrom \ \ \) \(\displaystyle \map f x\) \(\succcurlyeq_2\) \(\displaystyle \map f y\) Definition of Dual Ordering

$\blacksquare$


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