User:Abcxyz/Sandbox/Dedekind Completions of Archimedean Ordered Groups

Yes, I know to put singular nouns in the theorem title when this is put up. (The title of this page refers to its subject.) It's just that I haven't spent the time to come up with (or discuss) a suitable name for this theorem. Just be patient; I'll get to the matter in due course. --abcxyz (talk) 17:45, 20 January 2013 (UTC)

Definition:Archimedean Ordered Group
An ordered group $\left({G, *, \preceq}\right)$ is said to be Archimedean iff:
 * $\forall g \in G: \left({\exists h \in G: \forall n \in \N: g^n \preceq h}\right) \implies g \preceq e$

where:
 * $e$ denotes the identity element of $\left({G, *}\right)$
 * $g^n$ denotes the $n$th power of $g$ in $\left({G, *}\right)$

Theorem
Let $\left({G, *, \preceq}\right)$ be an Archimedean ordered group.

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

Then there exists a binary operation $\tilde *$ on $\tilde G$ such that:
 * $({1}): \quad \bigl({\tilde G, \tilde *, \tilde \preceq}\bigr)$ is an ordered group
 * $({2}): \quad \phi$ is a group homomorphism from $\left({G, *}\right)$ to $\bigl({\tilde G, \tilde *}\bigr)$

Proof
Let $\tilde *: \tilde G \times \tilde G \to \tilde G$ be the binary operation on $\tilde G$ defined as:
 * $\forall x, y \in \tilde G: x \mathbin{\tilde *} y = \sup {\bigl\{{\phi \left({g * h}\right): \phi \left({g}\right) \mathrel{\tilde \preceq} x, \, \phi \left({h}\right) \mathrel{\tilde \preceq} y}\bigr\}}$

The existence of $x \mathbin{\tilde *} y$ is justified by Characterization of Dedekind Completion.

Note that $\phi$ is a homomorphism from $\left({G, *}\right)$ to $\bigl({\tilde G, \tilde *}\bigr)$.

From Supremum of Subset, it follows that $\tilde \preceq$ is compatible with $\tilde *$.

It remains to show that $\bigl({\tilde G, \tilde *}\bigr)$ is a group.

We now prove that $\tilde *$ is associative.

Let $e$ denote the identity of $\left({G, *}\right)$.

Then $\phi \left({e}\right)$ is the identity of $\bigl({\tilde G, \tilde *}\bigr)$.

We now prove the existence of inverses in $\bigl({\tilde G, \tilde *}\bigr)$.