Finite left-distributive algebras and embedding algebras\endtitle

Abstract

We consider algebras with one binary operation $\cdot$ and one generator ({\it monogenic}) and satisfying the left distributive law $a\cdot (b\cdot c)=(a\cdot b)\cdot (a\cdot c)$. 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~$A_\infty$. Results of Laver and Steel assuming a strong large cardinal axiom imply that $A_\infty$ is free; it is open whether the freeness of $A_\infty$ 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 {\it embedding algebra}. Using this and results of the first author, we conclude that the freeness of $A_\infty$ is unprovable in primitive recursive arithmetic.