User:Dfeuer/Definition:Total Positive Cone

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

Let $P$ be a Positive Cone $(G,\circ)$.

Then $P$ is a total positive cone or weak total positive cone iff:
 * $P \cup P^{-1} = G$

That is, $P$ is a total positive cone for $G$ if $P$ is a subset of $G$ such that:
 * $x,y \in P \implies x \circ y \in P$
 * $x \circ y \in P \implies y \circ x \in P$
 * $P \cap P^{-1} = \{e\}$
 * $P \cup P^{-1} = G$