Addition of Order Types/Examples/Example Ordering on Integers

Examples of Addition of Order Types

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.

Proof

Consider the following mappings:

$i_\N: \N \to \Z_{\ge 0}: x \mapsto x$

$\phi: \N \to \Z_{<0}: x \mapsto -\paren {x + 1}$

These are seen to be order isomorphisms.

We have that:

$\struct {\N, \le} \oplus \struct {\N, \le} \cong \struct {\Z, \preccurlyeq}$

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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