Binomials with tetration

[Ramanujan]

"Binomials" with tetration

With the hyperoperations of addition, iterated addition (multiplcation), which distributes over addition, and iterated multiplication (exponentiation), which distributes over multiplcation, we can study how exponentiation fails to distribute over the operation which is two iterative operationsbelow it, addition...Now in extending to tetration, how would tetration fail to distribute over multiplication ? What is (xy) tetra squared? Is there a system of elegant patterns to study, or is it trivial, unsolvable, etc.?

 

[Mentallic]

I'm not quite sure what you're asking, but maybe this is along the right lines?
http://en.wikipedia.org/wiki/Knuth's_up-arrow_notation