Order Type Addition is Associative
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Theorem
Let $\alpha$, $\beta$ and $\gamma$ be order types of ordered sets.
Then:
- $\paren {\alpha + \beta} + \gamma = \alpha + \paren {\beta + \gamma}$
where $+$ denotes order type addition.
Proof
Let $\struct {S_1, \preccurlyeq_1}$, $\struct {S_2, \preccurlyeq_2}$ and $\struct {S_3, \preccurlyeq_3}$ be ordered structures such that:
\(\ds \map \ot {S_1, \preccurlyeq_1}\) | \(=\) | \(\ds \alpha\) | ||||||||||||
\(\ds \map \ot {S_2, \preccurlyeq_2}\) | \(=\) | \(\ds \beta\) | ||||||||||||
\(\ds \map \ot {S_3, \preccurlyeq_3}\) | \(=\) | \(\ds \gamma\) |
where $\ot$ denotes order type.
Thus by definition of order type we are required to show that:
- $\paren {\struct {S_1, \preccurlyeq_1} \oplus \struct {S_2, \preccurlyeq_2} } \oplus \struct {S_3, \preccurlyeq_3} \cong \struct {S_1, \preccurlyeq_1} \oplus \paren {\struct {S_2, \preccurlyeq_2} \oplus \struct {S_3, \preccurlyeq_3} }$
where:
- $\oplus$ denotes order sum
- $\cong$ denotes order isomorphism.
Let:
\(\ds \struct {T_a, \preccurlyeq_a}\) | \(:=\) | \(\ds \struct {S_1, \preccurlyeq_1} \oplus \struct {S_2, \preccurlyeq_2}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \struct {S_1 \times S_2, \preccurlyeq_a}\) | ||||||||||||
\(\ds \struct {T_b, \preccurlyeq_b}\) | \(:=\) | \(\ds \struct {S_2, \preccurlyeq_2} \oplus \struct {S_3, \preccurlyeq_3}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \struct {S_2 \times S_3, \preccurlyeq_b}\) |
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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 $34 \ \text {(a)}$