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:

\(\ds \struct {T_a, \preccurlyeq_a}\) \(:=\) \(\ds \struct {S_1, \preccurlyeq_1} \oplus \struct {S_2, \preccurlyeq_2}\)
\(\ds \) \(=\) \(\ds \struct {S_1 \times S_2, \preccurlyeq_a}\)
\(\ds \struct {T_b, \preccurlyeq_b}\) \(:=\) \(\ds \struct {S_2, \preccurlyeq_2} \oplus \struct {S_3, \preccurlyeq_3}\)
\(\ds \) \(=\) \(\ds \struct {S_2 \times S_3, \preccurlyeq_b}\)





Sources