The Laver table $A_{n}$ is the unique algebra $(\{1,...,2^{n}\},*)$ such that $x*1=x+1$ for $1\leq x<2^{n}$, $2^{n}*1=1$, and $x*(y*z)=(x*y)*(x*z)$.

Let's now replace the Laver table $A_{n}$ with a similar structure $F_{n}$ which I shall call a fake Laver table.

Let $F_{n}=(\{1,...,2^{n}\},*_{n})$ be the algebra for all $n$ such that each $*_{n}$ is a binary operation where

  1. $2^{n}*_{n}x=x$ for $1\leq x\leq n$

  2. $x*_{n}y=((x-2^{n-1})*_{n-1}y)+2^{n-1}$ for $2^{n-1}<x<2^{n},1\leq y\leq 2^{n-1}$

  3. $x*_{n}y=((x-2^{n-1})*_{n-1}(y-2^{n-1})$ for $2^{n-1}<x<2^{n},2^{n-1}<y$

  4. $x*_{n}1=x+1$ for $1\leq x<2^{n}$

  5. $x*_{n}y=x*_{n-1}y$ whenever $x\leq 2^{n-1},1<y\leq 2^{n-1}$ and $x*_{n}(y-1)=2^{n}$ or $x*_{n}(y-1)<2^{n-1}$.

  6. $x*_{n}y=(x*_{n-1}y)+2^{n-1}$ whenever $x\leq 2^{n-1},1<y\leq 2^{n-1}$ and $2^{n-1}\leq x*_{n}(y-1)<2^{n}$.

  7. $x*_{n}y=x*_{n}(y-2^{n-1})$ whenever $x\leq 2^{n-1}$ and $2^{n-1}<y$.

We have $A_{1}=F_{1},A_{2}=F_{2},A_{3}=F_{3}$, but $A_{n}\neq F_{n}$ but $A_{n}\neq F_{n}$ whenever $n>3$. Therefore, for $n>3$, the algebras $F_{n}$ are not self-distributive.

While the definition of the fake Laver tables is more complex than the definition of the Laver tables, the fake Laver tables have much simpler structure and they are easier to compute since the fake Laver tables are equivalent to the Sierpinski triangle. As far as I know, the fake Laver tables have not been studied by anyone anywhere.

Define $\ell_{n}=|\{(x,y,z)\in F_{n}^{3}|x*_{n}(y*_{n}z)=(x*_{n}y)*_{n}(x*_{n}z)\}|$.

Let $t_{n}=\frac{\ell_{n}}{8^{n}}$. Then $t_{n}$ is the probability that $x*_{n}(y*_{n}z)=(x*_{n}y)*_{n}(x*_{n}z)$ in $F_{n}$. The values $t_{n}$ are strictly decreasing.

Is there a recursive formula or a summation formula for $\ell_{n}$? Is $\lim_{n\rightarrow\infty}t_{n}=0$? If so, then how quickly does $t_{n}$ converge to $0$? If not, then what is $\lim_{n\rightarrow\infty}t_{n}=0$? Do the fake Laver tables $F_{n}$ satisfy any non-trivial identity which is a consequence of the self-distributivity law $x*(y*z)=(x*y)*(x*z)$?

Here are the first few values of $\ell_{n}$:

1, 8,64,512,4080,32304,253840,1978992,15310352,117588080, 897021712

Here are the first few values of $t_{n}$:

1 , 1 , 1 , 0.996094 , 0.98584 , 0.968323 , 0.943657 , 0.912568 , 0.876099, 0.835417.

The fake Laver tables in this question are different from the bad Laver tables I asked about in this question even though both of these classes of algebras exhibit a certain amount of self-distributivity.


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