# Order Type Addition is not Commutative

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## Theorem

The operation of order type addition is not commutative.

## Proof

Consider the order type $\omega^* + \omega$, where:

- $\omega$ denotes the order type of the natural numbers $\N$
- $\omega^*$ denotes the dual of $\omega$
- $+$ denotes addition of order types.

From Order Type of Integers under Usual Ordering, this is the order type of $\struct {\Z, \le}$, the set of integers under the usual ordering.

Now consider order type $\omega + \omega^*$.

From Ordered Set with Order Type of Natural Numbers plus Dual has Minimum Element, this is the order type of an ordered structure $\struct {S, \preccurlyeq}$ which has a smallest element.

But $\struct {\Z, \le}$ has no such smallest element.

It follows that $\struct {\Z, \le}$ is not isomorphic to $\struct {S, \preccurlyeq}$.

Hence $\omega + \omega^*$ and $\omega^* + \omega$ are different order types.

The result follows.

$\blacksquare$

## Sources

- 1996: Winfried Just and Martin Weese:
*Discovering Modern Set Theory. I: The Basics*... (previous) ... (next): Part $1$: Not Entirely Naive Set Theory: Chapter $2$: Partial Order Relations: Exercise $28 \ \text {(c)}$