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Show that the two ordered sets $(P,\leq)$ and $(Q,\leq)$ are order isomorphic if and only if there exist order preserving maps $$X: P\to Q \\ Y: Q \to P$$ such that $X\circ Y = \text{id}_Q$ and $Y\circ X = \text{id}_P$, $\text{id}$ being identify map.

Where and how do we use the identity map ?

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If f:X -> Y is an order isomorphism, so is its inverse g:Y -> X. Clearly f o g and g o f are identities. Conversely, if f is order preserving and f o g is identity, then, f is surjective, hence order isomorphism.

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