# Images of classical Laver tables with compatible order

The classical Laver tables can be given an ordering $\preceq$ such that if $y\preceq z$, then $x*y\preceq x*z$. Let $A_{n}=(\{1,…,2^{n}\},*_{n})$ denote the $n$-th classical Laver table. Let $\pi_{n}:A_{n}\rightarrow A_{n-1}$ be the homomorphism where $\pi_{n}(x)=x\,\textrm{Mod}2^{n-1}$. Let $\prec_{n}$ be the ordering on $\{1,…,2^{n}\}$ where $x\prec_{n}y$ if and only if $\pi_{n}(x)\prec_{n-1}\pi_{n}(y)$ or $\pi_{n}(x)=\pi_{n}(y)$ and $x < y$. Then there is a unique function $L:\{1,...,2^{n}\}\rightarrow\{1,...,2^{n}\}$ such that $x < y$ if and only if $x\prec y$. The function $L$ is an involution, i.e. $L(L(x))=x$ for all $x$. Define an operation $\#$ on $\{1,...,2^{n}\}$ by $x\#_{n}y=L(L(x)*_{n}L(y))$. In the following tables, the coordinates of the form $(i,i\#_{n}j)$ are white while all the other coordinates are colored black. Notice how the images converge (in the hyperspace topology) to a final image.
The 8×8 table.

The 16×16 table.

The 32×32 table.

The 64×64 table.

The 128×128 table.

The 256×256 table.

The 512×512 table.

The 1024×1024 table.

The following image is a 1024×1024 image the table for the $\sharp_{n}$ operation.