Order Isomorphism between Wosets is Unique

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Theorem

Let $\struct {S_1, \preceq_1}$ and $\struct {S_2, \preceq_2}$ be wosets.

Let $\struct {S_1, \preceq_1} \cong \struct {S_2, \preceq_2}$, that is, let $\struct {S_1, \preceq_1}$ and $\struct {S_2, \preceq_2}$ be order isomorphic.


Then there is exactly one mapping $f: S_1 \to S_2$ such that $f$ is an order isomorphism.


Proof

Let $f: S_1 \to S_2$ and $g: S_1 \to S_2$ both be order isomorphisms.

By Inverse of Order Isomorphism is Order Isomorphism, the inverse $f^{-1}$ is also an order isomorphism.

Let $h = f^{-1} \circ g$ be the composition of $f^{-1}$ and $g$, which, by Composite of Order Isomorphisms is Order Isomorphism, is itself an order isomorphism.

So, by Order Isomorphism from Woset onto Subset:

$\forall x \in S_1: x \preceq_1 \map h x$

Now we apply $f$ to $S_1$ and see that:

$\forall x \in S_1: \map f x \preceq_2 \map f {\map h x} = \map g x$


In a similar way we can show that:

$\forall x \in S_1: \map g x \preceq_2 \map f x$

Hence $f = g$ and the proof is complete.

$\blacksquare$


Sources

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