User:Dfeuer/Definition:Positive Cone
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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 if and only if:
- $P \cap P^{-1} = \{ e \}$
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.