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.
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.
), n = 0, 1, 2, 3, 4: 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.
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.