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 of algebraic and combinatorial interest. They occur in the study of racks and quandles.

Definition

For any nonnegative integer n, the n-th Laver table is the 2ⁿ × 2ⁿ table whose entry in the cell at row p and column q (1 ≤ p,q ≤ 2ⁿ) is defined as^[1]

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

where ${\displaystyle \star _{n}}$ is the unique binary operation that satisfies the following two equations for all p, q in {1,...,2ⁿ}:

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

(1)

and

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

(2)

Note: Equation (1) uses the notation ${\displaystyle x{\bmod {2}}^{n}}$ to mean the unique member of {1,...,2ⁿ} congruent to x modulo 2ⁿ.

Equation (2) is known as the (left) self-distributive law, and a set endowed with any binary operation satisfying this law is called a shelf. Thus, the n-th Laver table is just the multiplication table for the unique shelf ({1,...,2ⁿ}, ${\displaystyle \star _{n}}$) that satisfies Equation (1).

Examples: Following are the first five Laver tables,^[2] i.e. the multiplication tables for the shelves ({1,...,2ⁿ}, ${\displaystyle \star _{n}}$), n = 0, 1, 2, 3, 4:

${\displaystyle \star _{0}}$	1
1	1

${\displaystyle \star _{1}}$	1	2
1	2	2
2	1	2

${\displaystyle \star _{2}}$	1	2	3	4
1	2	4	2	4
2	3	4	3	4
3	4	4	4	4
4	1	2	3	4

${\displaystyle \star _{3}}$	1	2	3	4	5	6	7	8
1	2	4	6	8	2	4	6	8
2	3	4	7	8	3	4	7	8
3	4	8	4	8	4	8	4	8
4	5	6	7	8	5	6	7	8
5	6	8	6	8	6	8	6	8
6	7	8	7	8	7	8	7	8
7	8	8	8	8	8	8	8	8
8	1	2	3	4	5	6	7	8

${\displaystyle \star _{4}}$	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16
1	2	12	14	16	2	12	14	16	2	12	14	16	2	12	14	16
2	3	12	15	16	3	12	15	16	3	12	15	16	3	12	15	16
3	4	8	12	16	4	8	12	16	4	8	12	16	4	8	12	16
4	5	6	7	8	13	14	15	16	5	6	7	8	13	14	15	16
5	6	8	14	16	6	8	14	16	6	8	14	16	6	8	14	16
6	7	8	15	16	7	8	15	16	7	8	15	16	7	8	15	16
7	8	16	8	16	8	16	8	16	8	16	8	16	8	16	8	16
8	9	10	11	12	13	14	15	16	9	10	11	12	13	14	15	16
9	10	12	14	16	10	12	14	16	10	12	14	16	10	12	14	16
10	11	12	15	16	11	12	15	16	11	12	15	16	11	12	15	16
11	12	16	12	16	12	16	12	16	12	16	12	16	12	16	12	16
12	13	14	15	16	13	14	15	16	13	14	15	16	13	14	15	16
13	14	16	14	16	14	16	14	16	14	16	14	16	14	16	14	16
14	15	16	15	16	15	16	15	16	15	16	15	16	15	16	15	16
15	16	16	16	16	16	16	16	16	16	16	16	16	16	16	16	16
16	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16

There is no known closed-form expression to calculate the entries of a Laver table directly,^[3] but Patrick Dehornoy provides a simple algorithm for filling out Laver tables.^[4]

Properties

1. For all p, q in {1,...,2ⁿ}: ${\displaystyle \ \ 2^{n}\star _{n}q=q;\ \ p\star _{n}2^{n}=2^{n};\ \ (2^{n}-1)\star _{n}q=2^{n};\ \ p\star _{n}2^{n-1}=2^{n}{\text{ if }}p\neq 2^{n}}$.
2. For all p in {1,...,2ⁿ}: ${\displaystyle \ \ (p\star _{n}q)_{q=1,2,3,...}}$ is periodic with period π_n(p) equal to a power of two.
3. For all p in {1,...,2ⁿ}: ${\displaystyle \ \ (p\star _{n}q)_{q=1,2,3,...,\pi _{n}(p)}}$ is strictly increasing from ${\displaystyle p\star _{n}1=p+1\ }$ to ${\displaystyle \ p\star _{n}\pi _{n}(p)=2^{n}}$.
4. For all p,q: ${\displaystyle \ p\star _{n}q=(p+1)^{(q)},{\text{ where }}x^{(1)}=x,\ x^{(k+1)}=x^{(k)}\star _{n}x.}$^[1]

Are the first-row periods unbounded?

Looking at just the first row in the n-th Laver table, for n = 0, 1, 2, ..., the entries in each first row are seen to be periodic with a period that's always a power of two, as mentioned in Property 2 above. The first few periods are 1, 1, 2, 4, 4, 8, 8, 8, 8, 16, 16, ... (sequence A098820 in the OEIS). This sequence is nondecreasing, and in 1995 Richard Laver proved, under the assumption that there exists a rank-into-rank (a large cardinal property), that it actually increases without bound. (It is not known whether this is also provable in ZFC without the additional large-cardinal axiom.)^[5] In any case, it grows extremely slowly; Randall Dougherty showed that 32 cannot appear in this sequence (if it ever does) until n > A(9, A(8, A(8, 254))), where A denotes the Ackermann–Péter function.^[6]

References

1. Biane, Philippe (2019). "Laver tables and combinatorics". arXiv:1810.00548 [math.CO].
2. Dehornoy, Patrick (2014). "Two- and three-cocycles for Laver tables". arXiv:1401.2335 [math.KT].
3. Lebed, Victoria (2014), "Laver Tables: from Set Theory to Braid Theory", Annual Topology Symposium, Tohoku University, Japan (PDF). See slide 8/33.
4. Dehornoy, Patrick. Laver Tables (starting on slide 26). Retrieved 2018-12-11.
5. 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.
6. 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, S2CID 13242324.

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/