Should tetration be a multivalued function?

[marraco]

This gives a natural way to define °a for the Z branch. °a should be the value that makes:

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{^{-1}a\,^{^1a}\,=\,^1a\,\,^{^{-1}a}\,=\,^{0}a\,^{^0a}\, \; or \; (log_a{^0a})\,^{(a^{(^0a))}=(a^{(^0a)})\,^{(log_a{^0a})} \)

then it also satisfies:

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{(^{r+(n+1)}a)^{^{-(r+(n+1))}a}=(^{-(r+n)}a)^{^{r+n}a}} \)

I retract that. The last equation is satisfied no matter what the value of °a is.