Definition:Addition of Order Types

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

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 $\alpha + \beta$ be defined as:

$\alpha + \beta:= \map \ot {\struct {S_1, \preccurlyeq_1} \oplus \struct {S_2, \preccurlyeq_2} }$

where $\oplus$ denotes the order sum operator.

The operation $+$ is known as order type addition or addition of order types.


Example Ordering on Integers

Let $\preccurlyeq$ denote the relation on the set of integers $\Z$ defined as:

$a \preccurlyeq b$ if and only if $0 \le a \le b \text { or } b \le a < 0 \text { or } a < 0 \le b$

where $\le$ is the usual ordering on $\Z$.

Then the order type of $\struct {\Z, \preccurlyeq}$ is:

$\map \ot {\Z, \preccurlyeq} = \omega + \omega$

where $\omega$ denotes the order type of the natural numbers.