Zeration and Deltation

[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.