# Composite of Order Isomorphisms is Order Isomorphism

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## Theorem

Let $\struct {S_1, \preceq_1}$, $\struct {S_2, \preceq_2}$ and $\struct {S_3, \preceq_3}$ be ordered sets.

Let:

- $\phi: \struct {S_1, \preceq_1} \to \struct {S_2, \preceq_2}$

and:

- $\psi: \struct {S_2, \preceq_2} \to \struct {S_3, \preceq_3}$

Then $\psi \circ \phi: \struct {S_1, \preceq_1} \to \struct {S_3, \preceq_3}$ is also an order isomorphism.

## Proof

From Composite of Bijections is Bijection, $\psi \circ \phi$ is a bijection, as, by definition, an order isomorphism is also a bijection.

From Inverse of Composite Bijection, the inverse of $\psi \circ \phi$ is given by:

- $\paren {\psi \circ \phi}^{-1} = \phi^{-1} \circ \psi^{-1}$

By definition of composition of mappings:

- $\map {\psi \circ \phi} x = \map \psi {\map \phi x}$

By definition of order isomorphism, we have:

- $\phi: \struct {S_1, \preceq_1} \to \struct {S_2, \preceq_2}$ is an increasing mapping

and:

- $\psi: \struct {S_2, \preceq_2} \to \struct {S_3, \preceq_3}$ is an increasing mapping.

Hence from Composite of Increasing Mappings is Increasing:

- $\psi \circ \phi: \struct {S_1, \preceq_1} \to \struct {S_3, \preceq_3}$ is an increasing mapping.

Similarly by definition of order isomorphism:

- $\phi^{-1}: \struct {S_2, \preceq_2} \to \struct {S_1, \preceq_1}$ is an increasing mapping

and:

- $\psi^{-1}: \struct {S_3, \preceq_3} \to \struct {S_2, \preceq_2}$ is an increasing mapping.

Hence from Composite of Increasing Mappings is Increasing:

- $\phi^{-1} \circ \psi^{-1}: \struct {S_3, \preceq_3} \to \struct {S_1, \preceq_1}$ is an increasing mapping.

Hence we have that:

- $\psi \circ \phi: \struct {S_1, \preceq_1} \to \struct {S_3, \preceq_3}$ is an increasing mapping

and:

- $\paren {\psi \circ \phi}^{-1}: \struct {S_3, \preceq_3} \to \struct {S_1, \preceq_1}$ is an increasing mapping

and it follows by definition that $\psi \circ \phi$ is an order isomorphism.

$\blacksquare$

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 14$: Orderings: Theorem $14.1: \ 3^\circ$ - 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 14$: Orderings: Exercise $14.6$ - 1971: Gaisi Takeuti and Wilson M. Zaring:
*Introduction to Axiomatic Set Theory*: $\S 6.30$