Definition:Ordering Compatible with Ring Structure

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Let $\struct {R, +, \circ}$ be a ring whose zero is $0_R$.

An ordering $\preccurlyeq$ on $R$ is compatible with the ring structure of $R$ if and only if:

\((OR \, 1)\)   $:$   $\preccurlyeq$ is compatible with $+$:      \(\displaystyle \forall a, b, c \in R:\)    \(\displaystyle a \preccurlyeq b \)   \(\displaystyle \implies \)   \(\displaystyle \paren {a + c} \preccurlyeq \paren {b + c} \)             
\((OR \, 2)\)   $:$   Product of Positive Elements is Positive      \(\displaystyle \forall a, b \in R:\)    \(\displaystyle 0_R \preccurlyeq x, 0_R \preccurlyeq y \)   \(\displaystyle \implies \)   \(\displaystyle 0_R \preccurlyeq x \circ y \)             

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