Where can I learn more about commutative hyperoperations? 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.
 A: 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

A: 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.
