User:Abcxyz/Sandbox/Dedekind Completions of Ordered Sets

Definition:Supremum-Dense
Let $\left({S, \preceq}\right)$ be an ordered set.

Let $T \subseteq S$.

Then $T$ is supremum-dense in $\left({S, \preceq}\right)$ iff:
 * $\forall x \in S: x = \sup {\left\{{y \in T: y \preceq x}\right\}}$

The dual notion is infimum-dense.

Definition:Dedekind Completion
Let $S$ be an ordered set.

The Dedekind completion of $S$ consists of a Dedekind complete ordered set $\bigl({\tilde S, \preceq}\bigr)$ and an order embedding $\phi: S \to \tilde S$ such that:
 * $({1}): \quad$ The image $\phi \left({S}\right)$ is supremum-dense and infimum-dense in $\bigl({\tilde S, \preceq}\bigr)$.


 * $({2}): \quad \forall x \in \tilde S: \exists a, b \in \phi \left({S}\right): a \preceq x \preceq b$

Theorem
Let $S$ be an ordered set.

Let $\bigl({\tilde S, \le}\bigr)$ be a Dedekind complete ordered set.

Let $\phi: S \to \tilde S$ be an order embedding.

Then $\bigl({\bigl({\tilde S, \le}\bigr), \phi}\bigr)$ is the Dedekind completion of $S$ iff:
 * For all Dedekind complete ordered sets $\left({X, \preceq}\right)$, and all order embeddings $f: S \to X$, there exists an order embedding $\tilde f: \tilde S \to X$ such that $\tilde f \circ \phi = f$.

Necessary Condition
Let $\left({X, \preceq}\right)$ be a Dedekind complete ordered set.

Let $f: S \to X$ be an order embedding.

Let $\tilde f: \tilde S \to X$ be the increasing mapping defined as:
 * $\forall x \in \tilde S: \tilde f \left({x}\right) = \sup {\left\{{f \left({s}\right): \phi \left({s}\right) \le x}\right\}}$

The existence of $\tilde f \left({x}\right)$ is justified by the definition of the Dedekind completion.

We have that $\tilde f \circ \phi = f$.

Let $x, y \in \tilde S$ be such that $\tilde f \left({x}\right) \preceq \tilde f \left({y}\right)$.

Then:

Hence, $\tilde f$ is an order embedding.

Sufficient Condition
We show that:
 * $({1}): \quad$ The image $\phi \left({S}\right)$ is supremum-dense in $\bigl({\tilde S, \le}\bigr)$.


 * $({2}): \quad \forall x \in \tilde S: \exists b \in \phi \left({S}\right): x \le b$

The result will then follow from Dedekind Completeness is Self-Dual.

Define:
 * $T = \bigl\{{x \in \tilde S: x = \sup {\left\{{a \in \phi \left({S}\right): a \le x}\right\}}}\bigr.$ and $\bigl.{\exists b \in \phi \left({S}\right): x \le b}\bigr\}$

Then, by definition:
 * $({1}): \quad$ The image $\phi \left({S}\right)$ is supremum-dense in $\left({T, \le}\right)$.


 * $({2}): \quad \forall x \in T: \exists b \in \phi \left({S}\right): x \le b$

We show that $\left({T, \le}\right)$ is Dedekind complete.

Let $A \subseteq T$ be non-empty and bounded above in $\left({T, \le}\right)$.

Then $A$ admits a supremum in $\tilde S$, and, by Supremum of Suprema:
 * $\sup A = \sup {\left\{{a \in \phi \left({S}\right): \exists x \in A: a \le x}\right\}} \le \sup {\left\{{a \in \phi \left({S}\right): a \le \sup A}\right\}}$

It follows that $\sup A \in T$.

Hence, $\left({T, \le}\right)$ is Dedekind complete.

Let $\psi: S \to T$ denote the restriction of $\phi$ to $S \times T$.

By definition, there exists an order embedding $\epsilon: \tilde S \to T$ such that $\epsilon \circ \phi = \psi$.

It follows that:

Since $\epsilon$ is an injection, it follows that $T = \tilde S$.

Theorem
Let $S$ be an ordered set.

Let $\left({X, f}\right)$ and $\left({Y, g}\right)$ be Dedekind completions of $S$.

Then there exists a unique order isomorphism $\psi: X \to Y$ such that:
 * $\psi \circ f = g$

Proof
By Characterization of Dedekind Completion, there exist order embeddings $\tilde f: Y \to X$ and $\tilde g: X \to Y$ such that:
 * $\tilde f \circ g = f$
 * $\tilde g \circ f = g$

It follows that:
 * $\bigl({\tilde f \circ \tilde g}\bigr) \circ f = f$
 * $\bigl({\tilde g \circ \tilde f}\bigr) \circ g = g$

We have that:

Hence:
 * $\tilde f \circ \tilde g = \operatorname{id}_X$

Similarly:
 * $\tilde g \circ \tilde f = \operatorname{id}_Y$

Therefore, $\tilde g$ is an order isomorphism.

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

Then:
 * $\bigl({\psi \circ \tilde f}\bigr) \circ g = g$

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

Hence:
 * $\psi = \psi \circ \tilde f \circ \tilde g = \tilde g$

as desired.

Theorem
Let $\left({S, \preceq}\right)$ be an ordered set.

Let $C$ be a chain in $S$.

Let $C$ be supremum-dense in $S$.

Then $\preceq$ is a total ordering on $S$.

Proof
Let $x, y \in S$.

By Supremum of Subset, it suffices to show that one of:
 * $\left\{{a \in C: a \preceq x}\right\}$, $\left\{{a \in C: a \preceq y}\right\}$

is a subset of the other.

Suppose that:
 * $\left\{{a \in C: a \preceq y}\right\} \nsubseteq \left\{{a \in C: a \preceq x}\right\}$

Then there exists a $b \in C$ such that $b \preceq y$ and $b \npreceq x$.

Let $a \in C$, $a \preceq x$.

If $b \preceq a$, then, by the transitivity of $\preceq$, we have $b \preceq x$, contradicting the definition of $b$.

Hence, since $C$ is a chain, we have $a \preceq b$.

By the transitivity of $\preceq$, it follows that $a \preceq y$.

That is:
 * $\left\{{a \in C: a \preceq x}\right\} \subseteq \left\{{a \in C: a \preceq y}\right\}$

Hence the result.

Theorem
Let $S$ be a totally ordered set.

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

Then $\tilde S$ is totally ordered.

Proof
This follows directly from Ordered Set Containing Supremum-Dense Chain is Totally Ordered.