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.

Definition

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

L_n(p, q) := p \star q,

where p denotes the row and q denotes the column of the entry. Define

p \star 1 := p + 1 \mod 2^n

and then calculate the remaining entries of each row from the m-th to the first using the equation

p \star (q \star r) := (p \star q) \star (p \star r)

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 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. See the third page of the online preprint: "one does not know any closed formulas" for the multiplication law nor the row periods of the Laver tables.
  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, 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