User:Abcxyz/Sandbox/Dedekind Completions of Ordered Sets

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

Re: Dedekind completions
Let $S$ be an ordered set.

Let $\bigl({\bigl({\tilde{S}, \preceq}\bigr), \phi}\bigr)$ be a Dedekind completion of $S$.

Then:
 * $\forall a, b \in \tilde{S}: a \prec b \implies \exists x, y \in S: a \preceq \phi \left({x}\right) \prec b$, $a \prec \phi \left({y}\right) \preceq b$

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

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

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 $\psi = \tilde{g}$.