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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?

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    Why not call the sets $A_\alpha$ or $\alpha$?2012-03-16
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    Why not use something like $(A,B)$, where $A$ and $B$ are your sets? If you don't like parentheses, you can use brackets, or maybe you can use a bar $A|B$, $A*B$, or maybe just a comma, $A,B$ or a semicolon or just a colon without any braces whatsoever.2012-06-05

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