User:Abcxyz/Sandbox/Dedekind Completions of Ordered Sets

Feel free to comment. --abcxyz (talk) 16:44, 9 January 2013 (UTC)

Uniqueness of Dedekind Completion
Let $\phi: X \to X$ be an increasing mapping such that:
 * $\phi \circ f = f$

Then $\phi = \operatorname{id}_X$.

Suppose that $\psi: X \to Y$ is an order isomorphism such that:
 * $\psi \circ f = g$

By definition, there exists an order embedding $h: Y \to X$ such that:
 * $h \circ g = f$

It follows that:
 * $\psi \circ h = \tilde{g} \circ h = \operatorname{id}_Y$

Therefore, $h$ is an order isomorphism, and hence $\psi = \tilde{g}$.