User:Dfeuer/Definition:Positive Cone

Definition
Let $(G,\circ)$ be a group.

Let $P$ be a cone compatible with $(G,\circ)$.

Then $P$ is a positive cone or weak positive cone iff:
 * $P \cap P^{-1} = \{ e \}$

That is, if $P$ satisfies Cone Condition Equivalent to Antisymmetry and Cone Condition Equivalent to Reflexivity.

Theorem
Let $(G,\circ)$ be a group with identity $e$.

Let $P$ be a Positive Cone in $G$.

Let $\le$ be the relation induced by $P$.

Then $(G,\circ,\le)$ is an ordered group.

Proof
$\le$ is a Transitive Relation compatible with $\circ$ by User:Dfeuer/Cone Compatible with Group Induces Transitive Compatible Relation.

By the definition of a Positive Cone:
 * $P \cap P^{-1} = \{ e \}$

Thus
 * $P \cap P^{-1} \subseteq \{ e \}$
 * $e \in P \cap P^{-1}$

Thus $P$ satisfies both Cone Condition Equivalent to Antisymmetry and Cone Condition Equivalent to Reflexivity.

Thus $\le$ is transitive, antisymmetric, and reflexive, so it is an ordering.

Since $\le$ is an ordering compatible with $\circ$, $(G,\circ,\le)$ is an ordered group.

Theorem
Let $(G,\circ,\le)$ be an Ordered Group with identity $e$.

Let $P = {\bar\uparrow}e$.

Then $P$ is a Definition:Positive Cone inducing $\le$, and is the only positive cone to do so.

Proof
By the definition of an ordering, $\le$ is transitive, reflexive, and antisymmetric.

By the definition of an ordered group, $\le$ is compatible with $\circ$.

Thus by User:Dfeuer/Transitive Relation Compatible with Group Operation Induced by Unique Cone, $P$ is a unique cone inducing $\le$.

By Cone Condition Equivalent to Reflexivity:
 * $e \in P \cap P^{-1}$

By Cone Condition Equivalent to Antisymmetry:
 * $P \cap P^{-1} \subseteq \{ e \}$

Thus :$P \cap P^{-1} = \{ e \}$, so $P$ is a positive cone.