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I saw a line in my notes $A^{\alpha(\pi)}$ where $\alpha$ is defined to take values among the non-negative integers.

$A^2$ is all ordered pairs in $A$ and $A^3$ is the set of all the ordered triples. What is $A^0$? The set of functions defined on nowhere into $A$? Is it some kind of singleton set?

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Yes, it's the set of all functions from the empty set to $A$, of which there is exactly one, the empty function.

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