Ordered-field isomorphism

Let $F$ and $G$ be ordered fields (Ordered field). An ordered-field isomorphism $\varphi : F \to G$ is a bijective field homomorphism (Field homomorphism and embedding) that is order-preserving in both directions:

$a < b \;\Longleftrightarrow\; \varphi(a) < \varphi(b) \qquad \text{for all } a, b \in F.$

Two ordered fields are isomorphic if there exists an ordered-field isomorphism between them; we write $F \cong G$.

• Equivalently, $\varphi$ is a field isomorphism carrying the positive cone of $F$ onto that of $G$ ($\varphi(P_F) = P_G$); the inverse $\varphi^{-1}$ is then also an ordered-field isomorphism.
• Because a field homomorphism preserves all of $+, -, \cdot, {}^{-1}, 0, 1$, an ordered-field isomorphism identifies $F$ and $G$ as ordered fields completely: every field-theoretic and order-theoretic statement transfers across it.
• For homomorphisms out of a complete ordered field, order-preservation is automatic (Homomorphisms out of a complete ordered field are order-preserving); this is what makes the isomorphism in Uniqueness of the complete ordered field: $\mathbb{R}$ up to a unique isomorphism unique.