I have several sets $A_i$ and bijections between them. (As stated in my theorem) no composition of these bijections produces a permutation of $A_i$ not equal to identity. So every bijection is identified by the pair of sets between which it acts.
It would be to cumbersome to denote every bijection with a special letter (such as $\Phi$). I want to write it in some consise way.
For example I could denote the bijection from $A_i$ to $A_j$ as $\phi_{A_i,A_j}$ but this is not formally right as I would first to prove $A_i\ne A_j$. Then I would denote it $\phi_{i,j}$ but this way I would need explicitly number my sets, but I'd better to use English names or maybe letters to denote the sets not numbers.
The best solution I found insofar is to denote every set with some letter and denote my bijections as $\phi_{\alpha,\beta}$, where $\alpha$ and $\beta$ determine some sets. This solution is not ideal, because it would involve for each considered set two different notations to denote it: say $A_1$ and $\alpha$.
This is insofar the best solution I know. But maybe somebody may suggest me a better language to formulate my theorem?