Laver table

In mathematics, Laver tables (named after Richard Laver, who discovered them towards the end of the 1980s in connection with his works on set theory) are tables of numbers that have certain properties. They occur in the study of racks and quandles.

Definition

For a given natural number n, one can define the n-th Laver table (with 2ⁿ rows and columns) by setting

${\displaystyle L_{n}(p,q):=p\star q}$,

where p denotes the row and q denotes the column of the entry. The operation ${\displaystyle \star }$ is the unique operation that satisfies the equations

${\displaystyle p\star 1:=p+1\mod 2^{n}}$

and

${\displaystyle p\star (q\star r):=(p\star q)\star (p\star r)}$.

The latter is sometimes referred to as the self-distributive law, and sets satisfying only that property are called shelves.

The resulting table is then called the n-th Laver table; for example, for n = 2, we have:

	1	2	3	4
1	2	4	2	4
2	3	4	3	4
3	4	4	4	4
4	1	2	3	4

There is no known closed-form expression to calculate the entries of a Laver table directly.[1]

Periodicity

When looking at the first row of entries in a Laver table, it can be seen that the entries repeat with a certain periodicity m. This periodicity is always a power of 2; the first few periodicities are 1, 1, 2, 4, 4, 8, 8, 8, 8, 16, 16, ... (sequence A098820 in the OEIS). The sequence is increasing, and it was proved in 1995 by Richard Laver that under the assumption that there exists a rank-into-rank (a large cardinal), it actually increases without bound.[2] Nevertheless, it grows extremely slowly; Randall Dougherty showed that the first n for which the table entries' period can possibly be 32 is A(9,A(8,A(8,255))), where A denotes the Ackermann function.[3]

References

1. Lebed, Victoria (2014), "Laver Tables: from Set Theory to Braid Theory", Annual Topology Symposium, Tohoku University, Japan (PDF). See slide 8/33.
2. Laver, Richard (1995), "On the algebra of elementary embeddings of a rank into itself", Advances in Mathematics, 110 (2): 334–346, doi:10.1006/aima.1995.1014, hdl:10338.dmlcz/127328, MR 1317621.
3. Dougherty, Randall (1993), "Critical points in an algebra of elementary embeddings", Annals of Pure and Applied Logic, 65 (3): 211–241, arXiv:math.LO/9205202, doi:10.1016/0168-0072(93)90012-3, MR 1263319.

Further reading

• Dehornoy, Patrick (2001), "Das Unendliche als Quelle der Erkenntnis", Spektrum der Wissenschaft Spezial (1): 86–90.
• Dehornoy, Patrick (2004), "Diagrams colourings and applications" (PDF), Proceedings of the East Asian School of Knots, Links and Related Topics, pp. 37–64.
• Shelves and the infinite: https://johncarlosbaez.wordpress.com/2016/05/06/shelves-and-the-infinite/