Definition:Addition of Order Types
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Definition
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.
Examples
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.
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