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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?

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up vote 1 down vote accepted

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 $\gamma+\alpha<\gamma+\beta$.

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