I am currently trying to do some work based on someone else's paper, but I am having trouble understanding some of its concepts.
Let $N$ be a finite non-empty set, and let $m\in\mathbb{N}=\{1,2,\dots\}$ be a natural number. Then, set $A_m=\{a_1,a_2,\dots,a_m\}$. Given any natural number $m\in\mathbb{N}$, let $\mathcal{P}(A_m)$ be the set of all linear orders (i.e., complete, transitive, reflexive and antisymmetric binary relations) on the set $A_m$. Then, let \begin{gather} f:\bigcup_{m\in\mathbb{N}}\mathcal{P}(A_m)^N\to A_m \end{gather} be a function that, for every natural number $m\in\mathbb{N}$, associates each $n$-tuple of linear orders over $A_m$ with an element in the set $A_m$.
Fix any such function $f$, and consider any strict and non-empty subset $B\subsetneq A_m$. I am wondering whether I can define the restriction of $f$ to $\mathcal{P}(B)^N$. While I think I can, I am not sure; because I am having trouble verifying that $\mathcal{P}(B)^N$ is a subset of the domain of $f$.
Therefore, given any function $f:\bigcup_{m\in\mathbb{N}}\mathcal{P}(A_m)^N\to A_m$ and any set $B\subsetneq A_m$, is it correct to define the restriction of $f$ to $\mathcal{P}(B)^N$?