Zeration = inconsistant ?

[MphLee]

\( a[0]b:=max(a,b)+1 \) \( if a \neq b \)
\( a[0]b:=max(a,b)+2 \) \( if a = b \)

This is the Rubtsov's definition of Zeration.
It satisfies the distributivity.
Addition distributes over the addition (easy to see).

As I said before

Using a classic definition over the naturals (starting from the addition) leaves an opening for a non-trivial solution for zeration below the addition because if we try to define it starting from the higher Hos. (the addition) we have to use the cutoff subtraction \( {-}^{*} \) (not defined if \( a\lt n \) ).
\( a[0]n=(a+((a{-}^{*}n)+1)) \)
This means that for the recursive definition of the hyperoperation sequence on the naturals we have an infinite ammount of functions that "models" Zeration (some are commutative!) and each of those contains the successor function: in other worlds, those models should behave as the successor function over a restricted subset of \( \mathbb{N} \times \mathbb{N} \)).

the recursive definition of the hyperoperations over the naturals make us able to have alot of different solution for Zeration, the most interesting solutions are probably the ones definable via Max operator.

example:
Rubtsov's Zeration
\( a[0]_{RR}b:=max(a,b)+1 \) \( if a \neq b \)
\( a[0]_{RR}b:=max(a,b)+2 \) \( if a = b \)

Trappman's Zeration
\( a[0]_{HT}b:=max(a+2,b+1) \)

Both are possible models of Zeration inside the Rubtsov-Romerio Hyperoperations Sequence.

Inside the classic sequence we could use

\( a[0]_{Classic}b:=max(a,b)+1 \)

Those three solutions are related witht he max operator. We could call them "tropical solutions for Zeration" maybe.