Where can I learn more about commutative hyperoperations?

Solution 1:

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

Solution 2:

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.