Order Type Multiplication Distributes over Addition

Theorem

Let $\alpha$, $\beta$ and $\gamma$ be order types of ordered sets.

Then:

$\alpha \cdot \paren {\beta + \gamma} = \paren {\alpha \cdot \beta} + \paren {\alpha \cdot \gamma}$

where:

$+$ denotes order type addition

$\cdot$ denotes order type multiplication.

Proof

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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 $34 \ \text {(c)}$