Let $R=R_{i\in n}$ is a family of ordinals where $n$ is also an ordinal.
Let $q_i(x) = \sum_{j \in i} R_j + x$ for every $x\in R_i$.
Is $q_i$ an invertible function?
Let $R=R_{i\in n}$ is a family of ordinals where $n$ is also an ordinal.
Let $q_i(x) = \sum_{j \in i} R_j + x$ for every $x\in R_i$.
Is $q_i$ an invertible function?
I find your notation rather confusing, so I’m going to recast the question. As I understand it, you have a set $A=\{\alpha_\xi:\xi\in\eta\}$ or ordinals, where $\eta$ is also an ordinal. For $\xi\in\eta$ and $\beta\in\alpha_\xi$ you’ve defined $q_\xi(\beta)=\left(\sum_{\zeta\in\xi}\alpha_\zeta\right)+\beta\;,$ where all addition is ordinal addition, and you want to know whether $q_\xi$ is injective.
It is, because ordinal addition has the property that if $\alpha,\beta,\gamma$ are ordinals, and $\alpha<\beta$, then $\gamma+\alpha<\gamma+\beta$.