\begin{definition}[Definition:Ordering Induced by Positivity Property]
Let $\struct {R, +, \circ, \le}$ be an ordered ring whose zero is $0_R$ and whose unity is $1_R$.
Let $P \subseteq R$ such that:
:$(1): \quad P + P \subseteq P$
:$(2): \quad P \cap \paren {-P} = \set {0_R}$
:$(3): \quad P \circ P \subseteq P$
Then the ordering $\le$ compatible with the ring structure of $R$ is called the '''ordering induced by (the positivity property) $P$'''.
\end{definition}