# What is the largest Laver table which has been computed?

Richard Laver proved that there is a unique binary operation $*$ on $\{1,\ldots,2^n\}$ which satisfies $$a*1 \equiv a+1 \mod 2^n$$ $$a* (b* c) = (a* b) * (a * c).$$ This is the $n$th Laver table $(A_n,*)$.

There is an algorithm for computing $a * b$ in $A_n$, but in general (and especially for small values of $a$), this requires one to compute much of the rest of $A_n$. What is the largest value for $n$ for which someone can, in a modest amount of time, compute an arbitrary entry in $A_n$? I am able to compute entries in $A_{27}$.

I should note that the map which sends $a$ to $a\ \mathrm{mod}\ 2^m$ defines a homomorphism from $A_n$ to $A_m$ for $m < n$ and hence the problem becomes strictly harder for larger $n$.

Edit: I have actually been able to compute $A_{28}$, not just $A_{27}$.

• Do you have a reference you could give for the algorithm? Thanks, – Apollo Mar 9 '11 at 22:52
• @Apollo: The one I used to get to $A_{27}$ is based on ideas in Dehornoy's book "Braids and self distributivity" where he discusses the function $\theta$. The basic idea for computing in $A_n$ is given by the following identities: $a*k = (a+1)_{[k+1]}$ for $a < 2^n$ and $2^n * k = k$. Here $a_{[k]}$ is the $k$th left associated power of $a$. This allows you to start at the bottom of the table and work up. Implementing this directly allows me to get to $A_{19}$ (there are problems both with time and memory for $A_{20}$). Contact me offlist for code, if you like. – Justin Moore Mar 10 '11 at 0:00
• It might be that you can run the distribution "backwards". Do you know how to find a,b,and c, given m and n, such that ab = m and ac =n ? Also, can you give a reference for Laver's result that left self-distributive * is unique up to isomorphism? Gerhard "Ask Me About System Design" Paseman, 2011.03.11 – Gerhard Paseman Mar 11 '11 at 20:56
• @Gerhard: Dehornoy's book is perhaps the best reference for non set theorists. – Andrés E. Caicedo Mar 11 '11 at 21:08
• @Gerhard:Dehornoy's book is good for both set theorists and non set theorists. It won an award. Also read Laver's original papers in Advances in Math. (90's, I think). They are well written. Mostly they concern the algebra of elementary embeddings, but there is something at the end about the Laver tables. – Justin Moore Mar 12 '11 at 1:25

The largest classical Laver table computed is actually $$A_{48}$$. The 48th table was computed by Dougherty and the algorithm was originally described in Dougherty's paper here. With today's technology I could imagine that one could compute $$A_{96}$$ if one has access to a sufficiently powerful computer.
I've been in contact with Patrick Dehornoy and Ales Drapal and both thought that $A_{28}$ is likely the current record for a Laver table computation.