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.

A remedial public service announcement about the second law of thermodynamics.

So it appears that some people in the mathematical community have some trouble with the second law of thermodynamics since they do not believe in science. They instead believe in perpetual motion machines, the world is flat (or a torus, or a projective sphere), and that computers, refrigerators, TV’s, cars, and airplanes are powered by demons rather than gasoline and electricity. Let me therefore give you a few experiments that you can do at home on the second law of thermodynamics.

WARNING: All of these experiments are AT YOUR OWN RISK. I cannot be held responsible for any electric shocks, burns, other injuries, dumb patents, or any other damages as a result of these experiments.

A scientific experiment on perpetual motion machines

So let me give you a do-it-yourself experiment that you can do that will hopefully convince you that perpetual motion machines do not exist.

Step 1: Find an extension cord. Extension cords can typically be found attached to electrical outlets or around equipment that requires electricity such as televisions or computers. If you cannot find an extension cord, then you may be able to buy one at Wal-Mart or Staples.
Extension cord

Step 2: Plug the extension cord into itself.

Extension cord plugged into itself

Step 3: Turn the extension cord on and then plug other things into the extension cord.

Step 4: Does the light on the extension cord turn on and do the things plugged into the extension cord turn on and/or power up? If the light on the extension cord turns on and your appliances receive power, then, congratulations, you have created perpetual energy. Go and patent your invention and make trillions of dollars. If the light does not turn on, then perpetual energy does not exist, and the second law of thermodynamics is true.

An experiment on the second law of thermodynamics

Step 1: Make a cup of some hot coffee. Do not drink the coffee.

Coffee

Step 2: Measure the temperature of the coffee with a thermometer and the temperature of the room which the coffee is in. If you do not have a thermometer, then you may find one at Wal-Mart or some other store. The temperature of the coffee should be much hotter than the temperature of the room. Please make sure that you have a thermometer and not a moisture meter.

Thermometer

Step 3: Wait 24 hours. Do not move the coffee. Store the coffee in a regular cup. Do not store the coffee in a thermos. Make sure that your mother does not heat up your coffee again since that will ruin this experiment.

Mother

Step 4: Measure the temperature of the coffee and the temperature of the room which you have left the coffee in.

Step 5: If the temperature of the coffee is equal to the temperature of the room, then the second law of thermodynamics is correct.
If you find that the temperature of the coffee is still near a boiling temperature, then you may have been using a moisture meter instead of a thermometer. In this case, you should (at your own risk of course) stick your finger into the coffee. If you do not get burned, then the second law of thermodynamics is correct after all. If you do get burned, then you need to ask your mother if she heat up the coffee again. If she denies heating up your coffee again, then the second law of thermodynamics is false. You should therefore publish your results in the top physics journal. If they reject you, then they are persecuting you.

A computer experiment on the second law of thermodynamics

Here is another remedial experiment which you can perform to convince you that not only does the second law of thermodynamics apply to our physical universe, but the second law also applies to any time reversible universe.

Step 1: Click here

Step 2: Draw an image in the field. You may draw any image that you like.

image in cellular automata

Step 3: Select a rule or input a rule yourself. You may use any rule that you like. After you select a rule, then you should run the rule for at least 100,000 generations.

Step 4:

If the resulting image looks something like this,
image
then try running your cellular automaton under a different rule and repeat steps 1-4 a few times. If the image still looks orderly every time, the perhaps the second law of thermodynamics does not hold for all universes.

On the other hand, if the resulting image looks like this,

Image

then the second law of thermodynamics works not just in our physical universe but also in any hypothetical or simulated universe such as a cellular automaton as long as that universe is time reversible.

An experiment on the second law of thermodynamics

With this experiment, you will be able to tell whether the universe will spontaneously violate the second law of thermodynamics and make all of the oxygen molecules migrate to the other side of the room while the nitrogen molecules stay on your side of the room.

Step 1: Gather the materials. For this experiment, you will need an oxygen mask and a full tank of oxygen. Please make sure you know how to use the oxygen mask before starting this experiment. You should also bring a fidget spinner to entertain yourself since this experiment may take a long time to complete and you may get bored. Oh. And make sure to pack some lunch and dinner since this experiment may take a while.

Fidget spinner

oxygen

Step 2: Once you have the required materials, go to a room and lock the door. Please prepare to stay in that room for up to 10 hours since it may take a while for the second law of thermodynamics to be violated.

Step 3: If you feel like you cannot breathe, then all of the oxygen molecules have migrated to the other side of the room in violation of the second law of thermodynamics. In this scenario, immediately put on your oxygen mask since it may take a while for the violation of the second law of thermodynamics to wear off. If the second law of thermodynamics is still being violated as your oxygen tank is about to be depleted, then please leave the room and get some fresh air. At this point, you should probably determine whether this phenomenon is a true violation of the second law of thermodynamics or whether your house is haunted. In order to tell if your house is haunted, please get it inspected. Any good home inspector will be able to tell whether evil spirits live in your home or not. If you paid 40,000 dollars for a house that normally costs 500,000 dollars, then your house is probably haunted; I mean, why would anyone sell a non-haunted house for 1/10th of its worth?

Step 4: If nothing happens and you do not have to put your oxygen mask on after 10 hours, then the second law of thermodynamics is probably correct.

Satire notice: This post is satire. I wrote this post in order to address the anti-scientific beliefs which I have found among the mathematical community, namely the denial of Landauer’s principle.