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.

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    $\begingroup$ 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') $\endgroup$ – Hurkyl May 12 '13 at 20:25
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    $\begingroup$ 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. $\endgroup$ – Dave L. Renfro May 13 '13 at 21:37
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    $\begingroup$ @DaveL.Renfro, that's pretty cool. Where did you find it? $\endgroup$ – goblin May 13 '13 at 21:47
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    $\begingroup$ 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. $\endgroup$ – Dave L. Renfro May 13 '13 at 21:58
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    $\begingroup$ @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) $\endgroup$ – Dave L. Renfro May 14 '13 at 14:20

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 $i=1$


and we have that


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


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    $\begingroup$ sounds interesting, but that link at the top of your posts goes to a "Error 404. Document not found" page. $\endgroup$ – goblin May 13 '13 at 21:49
  • $\begingroup$ @user18921 updated the link, anyways if doesn not work, try several times to open it, or copy and paste the http. $\endgroup$ – MphLee May 14 '13 at 7:36
  • $\begingroup$ @user18921 Now can you see the book?(in the second link you must chose "BOOK_ENG.PDF" ) $\endgroup$ – MphLee May 14 '13 at 8:13
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    $\begingroup$ Okay, this link worked. Thank you. $\endgroup$ – goblin May 14 '13 at 11:49

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.


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