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I just learned about commutative hyperoperations, and they look interesting. However, the wikipedia page doesn't link to more information.

Is there an article or book where I can learn more? I'm especially interested in whether this is a "natural" sequence of definitions.

Note that the expression $k^{\log_k(a)+\log_k(b)}, k>0$ is independent of $k$, since it always equals $ab$. On the other hand, the expression $k^{\log_k(a)\log_k(b)}$ is dependent on $k$, and this makes me wonder whether there isn't a "better", more "natural" sequence out there.

goblin GONE
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    By this criteria, ordinary multiplication isn't natural, because 'constructing' it out of addition requires you to choose which real number will act as the multiplicative unit. (which we will henceforth call '1') –  May 12 '13 at 20:25
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    If anyone is interested, it's likely that Albert Bennet's 1915 paper was raised from obscurity when I came across it around 2001 or 2002 and sent Ioannis Galidakis a copy to include in his on-line bibliography on higher order operations, since Bennet's paper was overlooked by Knoebel in Knoebel's 1981 survey paper Exponentials Reiterated. – Dave L. Renfro May 13 '13 at 21:37
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    @DaveL.Renfro, that's pretty cool. Where did you find it? – goblin GONE May 13 '13 at 21:47
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    I came across it while going through journal volumes in a university library. There are quite a few journals (probably over 50) of which during the past 25 years or so I've flipped through every page of every volume, "data mining" interesting (to me) mathematical items. – Dave L. Renfro May 13 '13 at 21:58
  • @DaveL.Renfro, sounds like it eventually paid off! Have you managed to develop much of the theory of Bennet's operations? – goblin GONE May 13 '13 at 22:01
  • @DaveL.Renfro In your opinion why was unknown his work? Wasn't (or maybe still today) very interesting the topic of the hyperoperations? – MphLee May 14 '13 at 08:12
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    @MphLee: (and user18921) I've hardly looked at the paper, other than filing it away with a lot of literature I've accumulated on tetration and related topics, literature I sent copies of to Ioannis as I encountered it, at least up until around 2008 or 2009. This started when he expressed a lot of interest in this 9 September 1999 post, which was preceded by this 2 September 1999 sci.math post. Incidentally, one of the things I managed to eventually track down (continued) – Dave L. Renfro May 14 '13 at 14:20
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    (continuation) was the origin of the term "tetration". Around 2003 or 2004 I managed to determine Reuben Louis Goodstein was responsible for the term and sent Ioannis the relevant documentation. Interestingly, I believe I found this by accident, not by any kind of literature (Jstor, MR reviews, etc.) digital search. The reason Bennet's paper is largely unknown is probably the same as most any other paper published at that time is largely unknown -- no one looks at them, and the extreme specialization that exists in math. – Dave L. Renfro May 14 '13 at 14:28
  • I just noticed the sci.math posts from 1999 I cited in my 14 May 2013 comment above no longer work. Here are the corresponding posts from the google sci.math archive: "This started when he expressed a lot of interest in this 9 September 1999 post, which was preceded by this 2 September 1999 sci.math post." (I note the google version dates this last one as 3 Sep 1999.) – Dave L. Renfro Oct 29 '23 at 16:22

2 Answers2

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Try to check the chapter $1$ (from Pag $9$) of this book (New Mathematical Objects-C.A.Rubtsov) :

Here the autor creates a kind of generalization of the Albert Bennet's Hyperoperations. But what he does is much more general. He creates a procedure that he calls $ω$-reflection using a function of connection $f$.

In the book the only function he uses is $f(x):=k^x$ where $k\gt 1$ is called in the book "factor of image".

Then he defines an infinite hierarchy of "reflexive binary operations" that are homomorphic via $f$:

$x\circ_iy=x+y$ if $i=1$

$f(x)\circ_{i+1}f(y)=f(x\circ_iy)$

and we have that

$x\circ_{i+1}y=f(f^{\circ-1}(x)\circ_if^{\circ-1}(y))$

The first chapter the autor puts more attention on the homomorphic operation $\circ_{3}$ that is denoted by $\odot$ in the book and called "reflexive multiplication". Since he uses the exponentiation as function of connection, $\circ_{3}$ is isomophic to the mutiplication, then commutative and associative.

$k^x \odot k^y = k^{a \times b}$

In this chapter the autor builds an infinite numbers of what he calls reflexive functions, reflexive (homomorphic) binary operations , reflexive algebras (that are homomorphics via f) and other mathematical objects in this way:

let $F$ a bijection $F:\Bbb R \rightarrow \Bbb R$ (function of connection)

he calls $f'$ the reflexive image of the function of $f$ via $F$ if we have

$f'\circ F=F \circ f$

If you want to go deeper, on Tetration forum there is a topic where the possible propeties/evaluation of reflexive binary operations with non-integer indexes $i\le 2$ using tetration is discussed: Rational Operators

More informations:

$1$ The first autor is C.A.Rubtsov that with G.F.Romerio worked on the Hyperoperations topic:

(Ackermann's function and new arithematical operations-Rubtsov, Romerio-2004)

$2$ About Albert Bennet here a pdf about his commutative Hyperoperations:

(Note on an Operation of the Thrid Grade- Albert A.Bennet )

if the link is broken try these

https://web.archive.org/web/20210715091833/http://www.geocities.ws/rubcov/english/09.htm

https://web.archive.org/web/20210715091855/http://www.geocities.ws/rubcov/english/10.htm
MphLee
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This answer outlines an application for these operators: defining an infinite sequence of abelian group structures with an infinite sequence of triplets like (+, -, 0), (×, ÷, 1), etc.

ismael
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