Finite Left-Distributive Algebras and Embedding Algebras

✍ Scribed by Randall Dougherty; Thomas Jech

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 495 KB
• Volume: 130
• Category: Article
• ISSN: 0001-8708
• DOI: 10.1006/aima.1997.1655

✦ Synopsis

We consider algebras with one binary operation } and one generator (monogenic) and satisfying the left distributive law a } (b } c)=(a } b) } (a } c). One can define a sequence of finite left-distributive algebras A n , and then take a limit to get an infinite monogenic left-distributive algebra A . Results of Laver and Steel assuming a strong large cardinal axiom imply that A is free; it is open whether the freeness of A can be proved without the large cardinal assumption, or even in Peano arithmetic. The main result of this paper is the equivalence of this problem with the existence of a certain algebra of increasing functions on natural numbers, called an embedding algebra. Using this and results of the first author, we conclude that the freeness of A is unprovable in primitive recursive arithmetic.

1997 Academic Press

1. Introduction

We consider algebras with one binary operation and one generator (monogenic) and satisfying the left distributive law a } (b } c)=(a } b) } (a } c); in particular, we look for a representation of the free algebra.

The word problem for the free monogenic left-distributive algebra was solved by Laver [6] under the assumption of a large cardinal and subsequently by Dehornoy [4] without such an assumption. Laver's result uses elementary embeddings from V * into V * under the ``application'' operation } defined by j } k= :<* j(k & V : ). If there exists such an embedding j other than the identity, then the algebra A j generated by j is free.

When the embeddings in A j are restricted to an initial segment of V * , they form a finite monogenic left-distributive algebra [7], and these finite algebras can be described without reference to elementary embeddings. In fact, for every n there is a (unique) left-distributive operation V n on the set A$ n =[1, 2, ..., 2 n ] such that a V n 1=a+1 for all a<2 n and 2 n V n 1=1.