Definition:Isomorphism (Abstract Algebra)/Ordered Structure Isomorphism

Definition
An isomorphism from an ordered structure $\left({S, \circ, \preceq}\right)$ to another $\left({T, *, \preccurlyeq}\right)$ is a mapping $\phi: S \to T$ that is both:


 * An isomorphism, i.e. a bijective homomorphism, from the structure $\left({S, \circ}\right)$ to the structure $\left({T, *}\right)$
 * An order isomorphism from the poset $\left({S, \preceq}\right)$ to the poset $\left({T, \preccurlyeq}\right)$.

Linguistic Note
The word isomorphism comes from the Greek morphe (μορφή) meaning form or structure, with the prefix iso- meaning equal.

Thus isomorphism means equal structure.