Distributive "tetration", is this possible

[tommy1729]

I've considered those operators before, I don't think they are related to this operator however.


I think the best way to start with this operator, is by observing it's base changing properties.

If \( a\, \otimes_\mu\, f(a, b) = b \)

then
\( ^t a\, \otimes_\mu\, f(a,b) =\, ^t b \)

So by observing the original base change formula, we should be on our way to finding a possible solution.

I think if we want to disprove its existence we'd best look at this relation:
\( (^t a)\,\otimes_\mu\, b = \,^t(a\, \otimes_\mu\,b) \)

this seems like the most unlikely property for any operator to have.

yes , try a = sqrt(2) and t = oo

regards

tommy1729