Definition:Ordering Compatible with Ring Structure

From ProofWiki
Jump to navigation Jump to search


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 $\preccurlyeq$ satisies the ring compatible ordering axioms:

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

Also see