Algebras from very large infinities : threshold changes in Laver tables ordered interpretably

The n-th monogenic Laver table is the unique algebraic structure A_n=({1,...,2^n},*) where x*(y*z)=(x*y)*(x*z) and x*1=x+1 mod 2^n for al x,y,z. One may use the thresholds to compute the n+1-th Laver table from the n-th Laver table. Let *_n denote the n-th Laver table operation. Define o_n(x) to be the least natural number where x*_n 2^{o_n(x)}=2^n. If x is in {1,...,2^(n-1)-1}, then we define the threshold \theta_n(x) to be the integer y in {0,...,2^o_n(x)} where x*y\leq 2^(n-1) and x*(y+1)\geq 2^(n-1)+1 and set \theta_n(2^(n-1))=0. The values \theta_n allow us to compute larger Laver tables from smaller ones. In most cases, \theta_(n-1)(x)=\theta_n(x), so we only need to take note of the instances where \theta_(n-1)(x) differs from \theta_n(x). This is a visualization of those cases where \theta_(n-1)(x) differs from \theta_n(x). While this sort of threshold calculation allows us to compute monogenic Laver tables up to about the 28th or 29th Laver table on a typical desktop computer, and it allows us to store these Laver tables in a memory efficient format, we can do better. Randall Dougherty was able to compute the 48th Laver table using his ideas from the paper Critical points in an algebra of elementary embeddings II, and I was able to (kind of) compute the 768th Laver table by extending Dougherty's ideas.

In this visualization, I reordered the elements of the Laver table that is more aligned with the metric induced by the congruences on the Laver table. If a pair of elements are in a congruence close to the identity congruence, then those two elements are closer together on the visualization. We can see that with this reordering the threshold changes do not jump around so much.

The notion of a monogenic Laver table is not my own nor have I been the first person to do this calculation with monogenic Laver tables.

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