Order Type Addition is Well-Defined Operation

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Theorem

The addition operation on order types is well-defined.


Proof

Let $\struct {S_1, \preccurlyeq_1}$ and $\struct {S_2, \preccurlyeq_2}$ be ordered sets.

Let $\struct {T_1, \preccurlyeq_{1'} }$ and $\struct {T_2, \preccurlyeq_{2'} }$ be ordered sets such that:

$\struct {S_1, \preccurlyeq_1}$ is isomorphic to $\struct {T_1, \preccurlyeq_{1'} }$
$\struct {S_2, \preccurlyeq_2}$ is isomorphic to $\struct {T_2, \preccurlyeq_{2'} }$


Let $\alpha := \map \ot {S_1, \preccurlyeq_1}$ and $\beta := \map \ot {S_2, \preccurlyeq_2}$ denote the order types of $\struct {S_1, \preccurlyeq_1}$ and $\struct {S_2, \preccurlyeq_2}$ respectively.

Let $\alpha' := \map \ot {T_1, \preccurlyeq_{1'} }$ and $\beta' := \map \ot {T_2, \preccurlyeq_{2'} }$ denote the order types of $\struct {T_1, \preccurlyeq_{1'} }$ and $\struct {T_2, \preccurlyeq_{2'} }$ respectively.


It is required to show that:

$\alpha + \beta$

is the same as:

$\alpha' + \beta'$


We have that:

\(\ds \alpha + \beta\) \(=\) \(\ds \map \ot {\struct {S_1, \preccurlyeq_1} \oplus \struct {S_2, \preccurlyeq_2} }\)
\(\ds \) \(=\) \(\ds \map \ot {\struct {T_1, \preccurlyeq_{1'} } \oplus \struct {T_2, \preccurlyeq_{2'} } }\) Order Isomorphism is Preserved by Order Sum
\(\ds \) \(=\) \(\ds \alpha' + \beta'\)

$\blacksquare$


Sources

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