User:Dfeuer/Transitive Relation Compatible with Group Operation Induced by Unique Cone
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Theorem
Let $\struct {G, \circ}$ be a group with identity $e$.
Let $\RR$ be a transitive relation compatible with $\circ$.
Let $C = \set {x \in G: e \mathrel \RR x}$.
Then $C$ induces $\RR$ and is the only compatible cone to do so.
Proof
Existence
Suppose $a \mathrel \RR b$.
Then $a \circ a^{-1} \mathrel \RR b \circ a^{-1}$, so
- $e \mathrel \RR b \circ a^{-1}$
so $b \circ a^{-1} \in C$.
If $b \circ a^{-1} \in C$, then
- $e \mathrel \RR b \circ a^{-1}$
So $a \mathrel \RR b$.
Thus $C$ induces $\RR$.
$\Box$
Uniquelness
Suppose $D$ induces $\RR$.
Let $x \in C$.
Then $x \circ e^{-1} = x$, so $e \mathrel \RR x$.
Thus $x = x \circ e^{-1} \in D$.
Let $x \in D$.
Then $x \circ e^{-1} = x \in D$.
So $e \mathrel \RR x$, so $x \in C$.
$\blacksquare$