Is there a function space for tetration?

[Chenjesu]

I am fairly confident there are unifying principals that can underly any tetrate like their algorithmic identity, similar to the gamma function's functional properties which can be used to contract a space for tetration. Has there been any paper on a kind of function space at least for integer heights of tetration? It might be useful as an extension of Galois theory in abstract algebra to test whether a specific result lies within the space of tetrates, if such a space exists.