Generalizations of Laver tables is posted (140 pages)

The full version of the paper Generalizations of Laver tables is now posted.

In the paper, I have focused on building the general theory of Laver tables rather than solving a major problem with regards to the Laver tables. In fact, one should consider this paper as an account of “what everyone needs to know about Laver tables” rather than “solutions to problems about Laver tables.” This paper lays the foundations for future work on Laver tables. Since there is only one paper on the generalizations of Laver tables as of August 2017, an aspiring researcher currently does not have to go through many journal articles in order to further investigate these structures. I hope and expect that this paper on Laver tables will incite a broad interest on these structures among set theorists and non-set theorists, and that further investigation on these structures will be made possible by this paper.

Researching Laver tables

If you would like to investigate Laver tables, then please investigate the permutative LD-systems, multigenic Laver tables, and endomorphic Laver tables instead of simply the classical Laver tables. Very little work has been done on the classical Laver tables since the mid 1990’s. The classical Laver tables by themselves are a dead-end research direction unless one investigates more general classes of structures.

The most important avenue of further investigation will be to evaluate the security and improve the efficiency of the functional endomorphic Laver table based cryptosystems. Here are some ways in which one can directly improve functional endomorphic Laver table based cryptography.

  1. Try to break these cryptosystems.
  2. Compute $A_{96}$.
  3. Find compatible linear orderings on Laver-like LD-systems.
  4. Find new multigenic Laver tables and new Laver-like LD-systems.

It usually takes about 15 years from when a new public key cryptosystem is proposed for the public to gain confidence in such a cryptosystem. Furthermore, people will only gain confidence in a new public key cryptosystem if the mathematics behind such a cryptosystem is well-developed. Therefore, any meaningful investigation into large cardinals above hugeness and the Laver tables will indirectly improve the security of these new cryptosystems.

While people have hoped for a strong connection between knots and braids and Laver tables, the Laver tables so far have not produced any meaningful results about knots or braids that cannot be proven without Laver tables. The action of the positive braid monoid is essential for even the definition of the permutative LD-systems, so one may be able to apply the permutative LD-systems to investigating knots and braids or even apply knots and braids to investigating permutative LD-systems. However, I would regard any investigation into the application of Laver tables to knots and braids to be a risky endeavor since so far people have not been able to establish a deep connection between these two types of structures.

If you are a set theorist investigating the Laver tables and you are not sure if you will stay in academia for your entire career, then I recommend for you to work on something that requires extensive computer programming. This will greatly improve your job prospects if you ever leave academia for any reason. Besides, today nearly all respectable mathematicians need to also be reasonably proficient computer programmers. You do not want to be in academia trying to help students get real-world jobs when you do not yourself have the invaluable real-world skill of computer programming.

My future work

I will not be able to work on Laver-like algebras too much in the near future since I am currently preoccupied with my work on Nebula, the upcoming cryptocurrency which will incentivize the construction of the reversible computer. I am already behind on my work on Nebula since this paper has taken most of my time already, so I really need to work more on Nebula now. Since developing and maintaining a cryptocurrency is a full-time job, I will probably not be able to continue my investigations on Laver tables.

Slides from talk at BLAST 2017 and a rant on giving talks about the classical Laver tables

Here are my slides for my talk at the BLAST 2017 conference at Vanderbilt University in Nashville, Tennessee.

As a side note, I just noticed this other conference. All of the talks at that other conference on Laver tables are woefully outdated (i.e. 1995 or so). They only talk about the classical Laver tables. As an analogy, only talking about the classical Laver tables is like only talking about the cyclic groups of order $3^{n}$ and then claiming that they some how represent group theory as a whole. If you are going to give a talk about Laver tables or write a paper on the Laver tables, then please read the abridged version of my paper before you do so.

The classical Laver tables by themselves are a rather dead-end research area that have not been active within the last 20 years (one can probably try to analyze the fractal structure obtained from the classical Laver tables but such an analysis will probably be difficult and incremental). In order to advance further research in this area, one needs to consider the generalizations including Laver-like algebras, multigenic Laver tables, and functional endomorphic Laver tables. The classical Laver tables do not explain what the subalgebras of $\mathcal{E}_{\lambda}/\equiv^{\gamma}$ generated by multiple elements look like (one cannot even show that $\mathcal{E}_{\lambda}/\equiv^{\gamma}$ is locally finite without using the multigenic Laver tables). The classical Laver tables do not have any cryptographic applications. The classical Laver tables are just one sequence of structures, and it is hard to advance mathematics simply by looking at only one kind of structure with limited complexity. There is no reason at all to look at the classical Laver tables without looking at more general structures.

It is better to call the structures $A_{n}=(\{1,…,2^{n}\},*_{n})$ “classical Laver tables ” instead of simply “Laver tables.” There are other structures to consider.

How to give a classical Laver table talk.

The first step to giving a presentation on the classical Laver tables is to make sure you give your talk to the proper audience. The best audience to give a classical Laver table talk to is an audience of middle schoolers or maybe high schoolers (it is not that hard to fill out the multiplication table of a classical Laver table). Once you have your audience of middle schoolers present, you should get them to fill out an $8\times 8$ classical Laver table and then a $16\times 16$ classical Laver table. After they fill out the $16\times 16$ classical Laver table. And yes, middle schoolers are completely capable of filling out classical Laver tables. It is not that hard. After they are done filling out the tables, you can show them pictures that arise from the classical Laver tables on the projector and hint about how these objects come from infinity.