Appending an ordinal to a monogenic Laver table

A reduced Laver algebra is an algebra (X,*,1) where X satisfies the identities
x*(y*z)=(x*y)*(x*z),x*1=1,1*x=x and where if x_n\in X for all n, then there is some N where x_0*...*x_N=1 where the implied parentheses are grouped on the left so that a*b*c=(a*b)*c.

Since Laver algebras or algebras of rank-into-rank embeddings (or similar classes of algebras) are complicated mathematical structures, it would be sensible to compute or study the simplest Laver algebras. One of the simpler classes of Laver algebras consists of the monogenic Laver tables (or just Laver tables) which are the Laver algebras with one generator; for each n, there is precisely one monogenic Laver table with n+1 critical points, namely the unique algebra A_n=({1,...,2^n-1,2^n},*) where * satisfies the equations x*(y*z)=(x*y)*(x*z) and x*1=x+1 mod 2^n.

The Laver algebras (X,*,1) with 2 generators a,c where c*c=1 are the next simplest Laver algebras to look at and shall be called a monogenic Laver table with an ordinal appended. In this video, we also add the condition that if x\in\langle a\rangle and crit(x)=crit(y*c), then x=y*a=1 for simplicity.
The top row alternates between yellow and blue to indicate a change in a critical point. Each time there is a color change in the top row, we move to the next critical point. The critical points of elements in \langle a\rangle are denoted by a red column. The other critical points are of the form
crit(a_[i]*c) where if 1\leq j\leq i, then crit(a_[j]) is less than crit(a_[i]). In this case, we display the binary expansion of i as a column (observe that there are often several of these binary expansions associated with the same critical point, so the yellow and blue row is necessary to tell us about any change in critical point).

The notion of a monogenic Laver algebra with an ordinal appended is my own.


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