Definition:Ordered Ring Isomorphism

Definition
Let $\left({S, +, \circ, \preceq}\right)$ and $\left({T, \oplus, *, \preccurlyeq}\right)$ be ordered rings.

An ordered ring isomorphism from $\left({S, +, \circ, \preceq}\right)$ to $\left({T, \oplus, *, \preccurlyeq}\right)$ is a mapping $\phi: S \to T$ that is both:


 * $(1): \quad$ An ordered group isomorphism from the ordered group $\left({S, +, \preceq}\right)$ to the ordered group $\left({T, \oplus, \preccurlyeq}\right)$


 * $(2): \quad$ A semigroup isomorphism from the semigroup $\left({S, \circ}\right)$ to the semigroup $\left({T, *}\right)$.

Also see

 * Definition:Ordered Structure Isomorphism