Axiom:Ring Compatible Ordering Axioms
Jump to navigation
Jump to search
Definition
Let $\struct {R, +, \circ}$ be a ring whose zero is $0_R$.
Let $\preccurlyeq$ be an ordering $\preccurlyeq$ on $R$.
$\preccurlyeq$ is an ordering compatible with ring structure on $R$ if and only if $\preccurlyeq$ satisfies the 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 | \(\ds \forall a, b \in R:\) | \(\ds 0_R \preccurlyeq a, 0_R \preccurlyeq b \) | \(\ds \implies \) | \(\ds 0_R \preccurlyeq a \circ b \) |
These criteria are called the ring compatible ordering axioms.