Zeration and Deltation

[Rayanso]

Hi, I'm a new user and I wanted to make share with you some questions and stuff related to Zeration (rank 0 hyper-operation) and its inverse, Deltation.

On this forum I discovered GFR's post and I looked at the paper, and I also used the wayback machine to download an old paper from 1996 on hyper-operations from KAR's website.
If you are not familiar with Zeration and Deltation I suggest that you read the file that GFR attached in his post.

One question I was asking myself was what 0.5 * (△0) would be equal to, and if it could be reduced to a single number. In KAR's paper, this is resolved by introducing a new unit "j", like the imaginary unit for (-1)^(0.5). We then have 2*j =  △0. The operation 0.5 * (△0) also has 2 solutions: △j and j.

We still don't know what 0.35801 * (△0), or with any other arbitrary coefficient, would equal to. I made a formula for any real number as a coefficient for what this would be equal to.

I came up with x * (△0) = j*2*(2n+1)*x with n being an integer. I verified this formula with (-1)^a = b^(△0 * a)

Verification: for (-1)^(1/2) we have b^(△0 * 1/2) = {b^j ; b^(△j)}, so we must have b^j = i
b is a real number that is not zero.

For (-1)^(1/3) we know that there will be 3 solutions, therefore, △0 * 1/3 will have 3 solutions. These are: △0 ; 2/3 * j ; -2/3 * j. All other solutions can be reduced to one of these 3.

We also have an interesting rule that completes one that already exists:

a^0 = 1
a^(△0) = -1
a^j = i
a^(△j) = -i

This makes sense because the absolute value is 1 in every case, and the exponent is linked in some way to the number 0. 
If we have b = a*(△0), for any real a. c^b will always have an absolute value of 1. 

I also discovered the rule △j = -j, which I think KAR has not mentioned in his paper.

Proof:

△j + △j = 2*△j = △0
△j + (-j) = △0 + j - j = △0
(-j) + (-j) = -(2*j) = -△0 = △0

This is as far as I went and there are many gaps in the algebraic structure, KAR used a new operation below Zeration for defining ln(△a) and created new units for other operations, and he mentioned that there is an infinite amount of new sets of numbers. It would be interesting to see if this can be generalized in some way.


I still have unanswered questions about this algebraic structure:

- How can we extend Zeration to the complex set? What would (3*i) ° 4 be equal to? What about j ° 2 ?
- What is the proof for Zeration's commutativity? I saw somewhere on the forum that there was a proof for that but I couldn't find it.
- What about other hypothetical extensions I didn't mention?
- Have I made any mistakes? What did other people discover?


Rayanso

[MphLee]

This is as far as I went and there are many gaps in the algebraic structure, KAR used a new operation below Zeration for defining ln(△a) and created new units for other operations, and he mentioned that there is an infinite amount of new sets of numbers. It would be interesting to see if this can be generalized in some way.

The hypothetical theory of "Delta Fields," as Rubtsov calls them in his monograph, is considered essentially proven by Rubtsov and a few others, though it is widely ignored in general, including by the "hyperoperations enthusiasts" community.

The main issue lies in the language and style that Rubtsov employs. His book does not adhere to the rigorous standards of exposition typically seen in the field. While this alone doesn’t say much about the content of the book, it presents a significant obstacle to understanding.

Nonetheless, there is value to be found. What is needed to extract meaningful information from his work is the language of category theory, which is a real theory of objects and mappings. Using this lens, we can understand that the entire book focuses on chains of homomorphic images of algebraic structures. In general, it’s concerned with inducing diagrams of mathematical structures and studying the iterated image and preimage functorial constructions in various cases.

If one is able to grasp this, the big question arises: how is this related to hyperoperations? The answer is that it is related only to Bennett’s hyperoperations! This is the revelation.

However, Rubtsov claims that some of the structures emerging from this study are crucial for understanding the 0th-rank hyperoperation in the sense of Goodstein's definition. Specifically, he argues that a spectrum of new number systems arises, connected to the algebraic inverses of the negative rank (Goodstein’s) hyperoperations.

One might doubt that Rubtsov is conflating Bennett's and classical hyperoperations out of confusion or carelessness. However, there are paragraphs in the book that demonstrate Rubtsov is aware of the difference, and is suggesting a possible connection. While this link is not proven, there is something intriguing about it. For further exploration, you are encouraged to carefully read the post Zeration = inconsistent?

I still have unanswered questions about this algebraic structure:

- How can we extend Zeration to the complex set? What would (3*i) ° 4 be equal to? What about j ° 2 ?
- What is the proof for Zeration's commutativity? I saw somewhere on the forum that there was a proof for that but I couldn't find it.
- What about other hypothetical extensions I didn't mention?
- Have I made any mistakes? What did other people discover?


Rayanso

Rubtsov-Romerio's Zeration is commutative by definition.
I have plenty of references and interesting readings on this, but it’s mostly detective work, as no comprehensive papers or books on the topic have ever been written. I suggest you take a look at the link I provided and check out our dear user Natsugous' paper on negative ranks.

[MphLee]

ADDENDUM:

Here's the link to Natsugou's paper: (Tetration Forum 2023) Negative ranks hyperoperations (new pdf)

And here's more discussion about defining zeration and negative rank operations: (Tetration forum 2022)How could we define negative hyper operators?, also in that thread look for the reference to Tencer's post on MSE [2015 Tencer].