Laver Tables – A Direct Approach

Abstract

In this thesis we have a closer look at Laver tables. On the front page the heat map of the 8 × 8 Laver table is shown. The colors of the bottom row represent in ascending order the numbers 1 to 8, purple being 1 and red being 8. This table is the compo- sition table of the unique system satisfying some algebraic property, namely the left self-distributivity, under the following condition: the first column starts with 2 ascends to 8 and ends with 1. One might already notice that each row is strictly ascending until it arrives at red and then repeats. This general periodicity of the rows can be proven. Furthermore if we consider larger tables one can show that the period of a fixed row is non-decreasing. For all tables that have been computed so far the period of the first row does not exceed 16. Now a natural question comes to mind:

Does the period of the first row have an upper bound or is it unbounded?

Assuming some large cardinal hypothesis Laver showed that the period of each row tends to infinity. This can be looked up in [Deh00][Chapter XIII, Proposition 1.15 on page 576]. But whether this is provable in the setting of set theory remains an open question.

The thesis is split up into two parts:
In Chapter 2 we give a formal introduction to Laver tables and state some common

facts, which we shall use in the course of this thesis. In the last section we shall restate a proof of the fact that the period of the first row grows extremely slow compared to primitive recursive functions assuming that the period is unbounded.

In Chapter 3 we prove some theorems that help us construct a Laver table from preceding ones. But it is still unclear whether some part has to actually be calculated or whether the whole table can be reconstructed by preceding ones. However, with this new knowledge one can show that the mean period grows linearly. In the last section we prove an interesting fact about “steps” that enables us to give equivalent characterizations about period jumps up to 8.

Download the Paper

Adler (2016): Laver Tables – A Direct Approach. ETH Zurich. Masterthesis, FS 16, Supervisor: Lorenz Halbeisen.

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