Laver table

Example

For $n \ge 0$, the Laver table $A_n$ is the set $\set{1, 2, \dots, 2^n}$ with the unique binary operation $\star$ such that:

1. $p \star 1 = p + 1$ for $p < 2^n$, and $2^n \star 1 = 1$;
2. $p \star (q \star r) = (p \star q) \star (p \star r)$ for all $p, q, r$.

So $A_n$ is a shelf: clause 2 is left self-distributivity. The table can be computed row by row from the bottom, where row $2^n$ is $2^n \star q = q$. For $p < 2^n$ one uses $p \star (q + 1) = (p \star q) \star (p + 1)$, where $p \star q > p$ lies in a row already known.

For $n = 2$ this gives the table of $A_2$, with $p$ indexing rows and $q$ indexing columns:

Check of one instance of clause 2: $1 \star (2 \star 1) = 1 \star 3 = 2$, and $(1 \star 2) \star (1 \star 1) = 4 \star 2 = 2$. All $64$ instances of clause 2 hold in $A_2$. The shelf $A_2$ is not a rack, since the row of $1$ takes only the values $2$ and $4$.