Composite of Ordered Ring Isomorphisms is Ordered Ring Isomorphism

Theorem

Let $\struct {S_1, +_1, \circ_1, \preccurlyeq_1}, \struct {S_2, +_2, \circ_2, \preccurlyeq_2}, \struct {S_3, +_3, \circ_3, \preccurlyeq_3}$ be ordered rings.

Let $\phi: S_1 \to S_2$ and $\psi: S_2 \to S_3$ be ordered ring isomorphisms.

Then the composite mapping $\psi \circ \phi: S_1 \to S_3$ is also an ordered ring isomorphism.

Proof

From Composite of Order Isomorphisms is Order Isomorphism, $\psi \circ \phi: \struct {S_1, \preccurlyeq_1} \to \struct {S_3, \preccurlyeq_3}$ is an order isomorphism.

From Composite of Isomorphisms in Algebraic Structure is Isomorphism, $\psi \circ \phi$ is an algebraic structure isomorphism.

From Isomorphism Preserves Groups, it follows that $\psi \circ \phi$ is a group isomorphism from $\struct {S_1, +_1}$ to $\struct {S_3, +_3}$.

From Isomorphism Preserves Semigroups, it follows that $\psi \circ \phi$ is a semigroup isomorphism from $\struct {S_1, \circ_1}$ to $\struct {S_3, \circ_3}$.

Hence the result.

$\blacksquare$

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $23$. The Field of Rational Numbers