Tetration Forum

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Is obvius that the mathematical concept of sequence of objects is formalized with the concept of indexed family (inside a set theory)

But my question is about who generalized the concept of Hyperoperation from the usual Goodstein 3-ary function (or the Knuth's uparrows) to some indexed families that satisfies some properties.

I'm tryng to continue the discussion started in the thread about the dustributive property of Bennet's operation family:

(05/27/2014, 08:22 PM)MphLee Wrote: [ -> ]
(05/27/2014, 07:45 PM)andydude Wrote: [ -> ]@MphLee

Hyperoperations, in the general sense, are any sequence of binary operations that includes addition and multiplication. The commutative hyperoperations satisfy this property because $\exp^0(\ln^0(a) + \ln^0(b)) = a + b$ and $\exp^1(\ln^1(a) + \ln^1(b)) = e^{\ln(a) + \ln(b)} = e^{\ln(a)}e^{\ln(b)} = a \times b$. That formula is the starting point, it is the definition of commutative hyperoperations. The fact that it contains addition and multiplication can be discussed and proved from the definition.

I'm even aware that the term Hyperoperations usually means (can be formalized as) an indexed family of binary operations $\{*_i\}_{i \in I}$ whith addition, multiplication and exponentiation belonging to the image of the indexed family (the image of the family is defined to be the image of the set of indexes- set of ranks- via the indicization function).

This definition is the one I found on Wikipedia and is very smart even if it cuts the Commutative hyperoperations out of the game (Maybe we can make a weaker concept of Hyperoperations Family without the exponentiation requirement, I would call them Weak Hyperoperations Families)...

Anyways I'm very courious...I was not able to find references about this terminology and I did not even find who introduced this formal definition.

Who actually gave the first definition of when an indexed family of binary operations is an Hyperoperations family? I need the reference because I made some improvement in the definition while writing a paper about the Hyperoperations.
First of all, I believe I wrote most of the Hyperoperations article on Wikipedia. There was an existing article called "Hyper operator", but there were many people on the Talk sub-page that led me to believe that the page needed a lot of work, so I tried to do my best with the rewrite. I included every reference that I could find on the topic, and compiled what I think is a comprehensive list of references that doesn't focus on Tetration, but all hyperoperations in general.

(07/17/2014, 05:29 PM)andydude Wrote: [ -> ]First of all, I believe I wrote most of the Hyperoperations article on Wikipedia. There was an existing article called "Hyper operator", but there were many people on the Talk sub-page that led me to believe that the page needed a lot of work, so I tried to do my best with the rewrite. I included every reference that I could find on the topic, and compiled what I think is a comprehensive list of references that doesn't focus on Tetration, but all hyperoperations in general.

Speaking of Wikipedia, has anyone else noticed that the "linear" approximation for tetration that is listed there is the naive version on the interval [-1,0], which is only C1 continuous for base e? I'm pretty sure we've discussed a better linear approximation somewhere, which is linear on a base-specific unit interval, which is C1 continuous, not just C0 continuous.

I stumbled upon it way back in the day:

I see the same issue with the wikepedia article for the superlogarithm:
http://en.wikipedia.org/wiki/Super-logar...roximation

In fact, that article explicitly calls out that it's C0 continuous, when a proper choice of interval would be C1 continuous. I found a post here on the forum where I discussed the C1 linear approximation of the slog:
http://math.eretrandre.org/tetrationforu...php?tid=98

Granted, these methods only work for real bases greater than eta*. However, I still think they're far more useful than the naive linear approximation.

* Real, because you need to be able to place e in a particular unit interval, based on iterated logarithms. Greater than eta, because, as I mentioned elsewhere, the linear approximation for bases less than eta is only valid between the primary fixed points, not between 0 and the lower fixed point.
(07/17/2014, 05:29 PM)andydude Wrote: [ -> ]First of all, I believe I wrote most of the Hyperoperations article on Wikipedia. There was an existing article called "Hyper operator", but there were many people on the Talk sub-page that led me to believe that the page needed a lot of work, so I tried to do my best with the rewrite. I included every reference that I could find on the topic, and compiled what I think is a comprehensive list of references that doesn't focus on Tetration, but all hyperoperations in general.