User:Dfeuer/Definition:Strict Total Positive Cone
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Definition
Let $(G,\circ)$ be a group with identity $e$.
Let $P$ be a Positive Cone $(G,\circ)$.
Then $P$ is a strict total positive cone iff:
- $P \cup P^{-1} \cup \{e\}= G$
That is, $P$ is a strict total positive cone for $G$ if $P$ is a subset of $G$ such that:
- $x,y \in P \implies x \circ y \in P$
- $x \circ y \in P \implies y \circ x \in P$
- $P \cap P^{-1} = \varnothing$
- $P \cup P^{-1} \cup \{ e \}= G$