How should I most clearly define the function that $f:\mathcal P_{fin}(\omega^{<\omega})\to \mathcal P_{fin}(\omega^{<\omega})$ which maps certain subsets of the power set of $\omega^{<\omega}$ to singletons, by deleting the smallest term of any ordinal (written in Cantor normal form) and reducing the exponent of all other terms by one:
$f:\displaystyle \bigcup_{s_1\in X}\{\omega^{b-1}\cdot s_1+\omega^{b-2}\cdot s_2+\ldots s_b:s_b \in X\subseteq\Bbb N\}\mapsto \{\omega^{b-2}\cdot s_1+\omega^{b-3}\cdot s_2+\ldots s_{b-1}\}$
Where $X\subseteq N$ and $s_b$ are drawn from $\Bbb N$?
Then this is surjective over $\mathcal P_{fin}(\omega^{<\omega})$ which indicates the subset of the power set, containing only finite sets of finite strings of naturals.
If this isn't clear, how best do I write this function so as to be clear?
Is "contraction mapping" an appropriate name for this function given that it must eventually terminate?
Best Answer
In your written definition of the function, there are several things that are unclear; for example, I don't really understand the emphasis on $s_1$ over the other terms, or how you're interpreting "$\bigcup$." I also don't understand your claim that it maps sets to singletons, given your example $\{\omega+1,\omega+2,\omega+3,\omega^2\cdot 3\}\mapsto\{1,\omega\cdot 3\}$.
Based on your example in the comments, though, I suspect that what you're asking for is the following. I'll first build it in stages, rather than describing it all at once, and then give an all-at-once definition below.
Incidentally, it does not always terminate; e.g. $f(\{\omega^n:n\in\mathbb{N}\})=\{\omega^n: n\in\mathbb{N}\}$ if I understand it correctly. Maybe you want to look at $\mathcal{P}_{fin}(\omega^{<\omega})$ - the set of finite sets of finite strings of naturals - instead of $\mathcal{P}(\omega^{<\omega})$? Also, the term "contraction mapping" has a technical meaning in the study of metric spaces, which this could conceivably fit but only after you've defined an appropriate metric structure.
First, let $C: \omega^{<\omega}\rightarrow\omega^\omega$ be the Cantor normal form function, sending a finite string of naturals to the ordinal $<\omega^\omega$ which it represents. (Note that $C$ is noninjective - for example, it sends both $\langle 0,1\rangle$ and $\langle 1\rangle$ to $1$ - so if you want an injection, it will be a slight modification of the obvious $C$.)
Next, we have the truncation function $$trun:\omega^{<\omega}\setminus\{\langle\rangle\}\rightarrow\omega^{<\omega}: \langle a_1,..., a_n\rangle\mapsto \langle a_1,..., a_{n-1}\rangle.$$ Note that this isn't defined on the empty string; if you prefer, you could define it to be the identity on the empty string, that is, set $trun(\langle\rangle)=\langle\rangle$.
Next, we have the pointwise version of your $f$: $$g:\omega^{<\omega}\rightarrow\omega^{<\omega}: C(\sigma)\mapsto C(trunc(\sigma)).$$ This does what you want on a single finite string of ordinals; e.g. $$g(\omega^2+\omega)=g(C\langle1,1,0\rangle)=C(trunc(\langle 1,1,0\rangle))=\omega+1.$$ Note that we have to show that $g$ is actually well-defined - that is, if $C(\alpha)=C(\beta)$ then $C(trunc(\alpha))=C(trunc(\beta))$ - but this is easy.
Finally, your $f$ is the "set version" of $g$: $$f:\mathcal{P}(\omega^{<\omega})\rightarrow\mathcal{P}(\omega^{<\omega}): S\mapsto\{g(s): s\in S\}.$$
OK, now here's the all-at-once definition: $f$ is the function from $\mathcal{P}(\omega^{<\omega})$ to $\mathcal{P}(\omega^{<\omega})$ which sends a set $X$ to the set
$$\{\omega^n\cdot s_{n+1}+\omega^{n-1}\cdot s_n+...+\omega^0\cdot s_1: \exists t(\omega^{n+1}\cdot s_{n+1}+\omega^{n}\cdot s_n+...+\omega^{1}\cdot s_1+t\in X)\}.$$