You are right, Trappmann solution isn't commutative, but following the original Zeration Thread we can see that Trapmann did propose that solution in order to outline the non-uniqueness of the Rubtsov solution: there is a non-commutative solution too!

By the way he later proposed an extension to the complex too, but i don't find this really interesting.

The real questions should be:

A-Why

A1-Why, between the infinite solutions for Zeration, we should chose the non-trivial ones?

A2-Is it an ideological and arbitrary choice?

A3-Is the request of commutativity a sufficently strong push?

A4-Is the existence of solutions definable via max operator a good reason for chosing a non-trivial solution?

Comment: the superfunction problem seems to suggest that the solution should be trivial (the successor)...but the quest for a commutative solution seems enough interesting and the tropical solutions give something interesting imho. Note also that, even if the idempotency of max implies that if you iterate it nothing happens, the Rubtsov's and the Trapmann's solutions are subfunctions of the addition even if over a restricted domain (because they behave as the successor on that restricted domain). If we use tropical solutions we have an usefull tool: the Litinov-Maslov dequantization can be used to reach, imho, the zerative inverses through a limit process (im still thinking about this).

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B-How (Uniqueness)

B1-Once we have chosen to go for a non-trivial solution to zeration, the Trappmann solution remind us that there is not uniqueness... how we can chose additional must-have properties in order to reach uniqueness?

B2-Do we really need associativity, commutativity and/or distributivity (higher ranks don't have them)?

Comment:the distributivity, as GFR said, doesn't hold for the higher ranks thus the requirement of distributivity (and associativity and commutativity) of hyperoperations lower then addition seems like something we want but probably isn't needed at all. Anyways is interesting to see that the Rubtsov's solution make us able to define a semifiels where the semifield addition is the RR-Zeration and the semifield multiplication is the common addition.

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If we want the solution of Zeration to satisfies some additional properties we have to work on the definition of the family: like adding "axioms" or using new recursion formulas. But if we do it we are talking about a totally different Hyperoperations family (or an exension):is like if we are "changing theory" (ex. from semirings to rings) This bring us to the last problem:

C-How (Hyperoperations sequences)

C1-How we should modify the definition of Hyperoperations?

C2-Once we have a great ammount of different Hyperoperations sequences definitions (i call them theories) wich one is the "real" one?

Comment:from a formalistic point of view there is not a real one. From a more platonist point of view those families are all real, we should just investigate their relationships.

These are hard questions...but maybe only for who doesn't like trivial solutions .

By the way he later proposed an extension to the complex too, but i don't find this really interesting.

The real questions should be:

A-Why

A1-Why, between the infinite solutions for Zeration, we should chose the non-trivial ones?

A2-Is it an ideological and arbitrary choice?

A3-Is the request of commutativity a sufficently strong push?

A4-Is the existence of solutions definable via max operator a good reason for chosing a non-trivial solution?

Comment: the superfunction problem seems to suggest that the solution should be trivial (the successor)...but the quest for a commutative solution seems enough interesting and the tropical solutions give something interesting imho. Note also that, even if the idempotency of max implies that if you iterate it nothing happens, the Rubtsov's and the Trapmann's solutions are subfunctions of the addition even if over a restricted domain (because they behave as the successor on that restricted domain). If we use tropical solutions we have an usefull tool: the Litinov-Maslov dequantization can be used to reach, imho, the zerative inverses through a limit process (im still thinking about this).

-------------------------

B-How (Uniqueness)

B1-Once we have chosen to go for a non-trivial solution to zeration, the Trappmann solution remind us that there is not uniqueness... how we can chose additional must-have properties in order to reach uniqueness?

B2-Do we really need associativity, commutativity and/or distributivity (higher ranks don't have them)?

Comment:the distributivity, as GFR said, doesn't hold for the higher ranks thus the requirement of distributivity (and associativity and commutativity) of hyperoperations lower then addition seems like something we want but probably isn't needed at all. Anyways is interesting to see that the Rubtsov's solution make us able to define a semifiels where the semifield addition is the RR-Zeration and the semifield multiplication is the common addition.

-------------------------

If we want the solution of Zeration to satisfies some additional properties we have to work on the definition of the family: like adding "axioms" or using new recursion formulas. But if we do it we are talking about a totally different Hyperoperations family (or an exension):is like if we are "changing theory" (ex. from semirings to rings) This bring us to the last problem:

C-How (Hyperoperations sequences)

C1-How we should modify the definition of Hyperoperations?

C2-Once we have a great ammount of different Hyperoperations sequences definitions (i call them theories) wich one is the "real" one?

Comment:from a formalistic point of view there is not a real one. From a more platonist point of view those families are all real, we should just investigate their relationships.

These are hard questions...but maybe only for who doesn't like trivial solutions .