Characterization of Strictly Increasing Mapping on Woset
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Lemma
Let $J$ and $E$ be well-ordered sets.
Let $h: J \to E$ be a mapping.
Let $S_\alpha$ denote an initial segment determined by $\alpha$.
The following are equivalent:
- $(1):\quad$ $h$ is strictly increasing and its image is either all of $E$ or an initial segment of $E$
- $(2):\quad$ $\forall \alpha \in J: h\left({\alpha}\right) = \min \left({E\setminus h\left[{S_\alpha}\right]}\right)$, and $h[S_\alpha] = S_{h(\alpha)}$
where:
- $h\left[{S_\alpha}\right]$ denotes the image of $S_\alpha$ under $h$
- $\min$ denotes the smallest element of the set.
Proof
$(1)$ implies $(2)$
Suppose $h$ satisfies:
- $h$ is strictly increasing and its image is either all of $E$ or an initial segment of $E$
Then for any $x,y \in J$:
| \(\ds x\) | \(\prec\) | \(\ds y\) | ||||||||||||
| \(\ds \implies \ \ \) | \(\ds h(x)\) | \(\prec\) | \(\ds h(y)\) | Definition of strictly increasing | ||||||||||
| \(\ds h[S_y]\) | \(=\) | \(\ds \left\{ { h(x) \in E: \exists x \in J: h(x) \prec h(y) } \right\}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds S_{h(y)}\) | ||||||||||||
| \(\ds \min\left({E \setminus h\left[{S_y}\right] }\right)\) | \(=\) | \(\ds \min\left({E \setminus S_{h(y)} }\right)\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds h(y)\) | Definition of smallest and of initial segment |
$\Box$
$(2)$ implies $(1)$
Suppose $h$ satisfies:
- $h(\alpha) = \min\left({E \setminus h\left[{S_\alpha}\right] }\right)$
By the Principle of Recursive Definition for Well-Ordered Sets, $h$ is thus uniquely determined.
Then:
| \(\ds h(y)\) | \(=\) | \(\ds \min\left({E \setminus h\left[{S_y}\right] }\right)\) | ||||||||||||
| \(\ds h[S_y]\) | \(=\) | \(\ds \left\{ { h(x) \in E: \exists x \in J: h(x) = \min\left({E \setminus h\left[{S_y }\right] }\right)} \right\}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \left\{ { h(x) \in E: \exists x \in J: h(x) \prec h(y)} \right\}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds S_{h(y)}\) | Definition of initial segment |
Thus for every $x \in S_y$, we have that $h(x) \in S_{h(y)}$.
Therefore $h$ is an strictly increasing mapping.
Furthermore, the image set of $h$ is the union of initial segments in $E$.
By Union of Initial Segments is Initial Segment or All of Woset, $h[J]$ is an initial segment of $E$ or all of $E$.
$\blacksquare$
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Sources
- 2000: James R. Munkres: Topology (2nd ed.): Supplementary Exercise $1.2$