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ISSN 1079-6762



Left-distributive embedding algebras

Authors: Randall Dougherty and Thomas Jech
Journal: Electron. Res. Announc. Amer. Math. Soc. 3 (1997), 28-37
MSC (1991): Primary 20N02; Secondary 03E55, 08B20
Published electronically: April 9, 1997
MathSciNet review: 1445632
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Abstract: We consider algebras with one binary operation $\cdot $ and one generator, satisfying the left distributive law $a\cdot (b\cdot c)=(a\cdot b)\cdot (a\cdot c)$; such algebras have been shown to have surprising connections with set-theoretic large cardinals and with braid groups. One can construct a sequence of finite left-distributive algebras $A_{n}$, and then take a limit to get an infinite left-distributive algebra $A_{\infty }$ on one generator. 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 left-distributive algebra of increasing functions on natural numbers, called an embedding algebra, which emulates some properties of functions on the large cardinal. Using this and results of the first author, we conclude that the freeness of $A_{\infty }$ is unprovable in primitive recursive arithmetic.

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Additional Information

Randall Dougherty
Affiliation: Department of Mathematics, Ohio State University, Columbus, OH 43210

Thomas Jech
Affiliation: Pennsylvania State University, 215 McAllister Building, University Park, PA 16802

Keywords: Left-distributive algebras, elementary embeddings, critical points, large cardinals, primitive recursive arithmetic
Received by editor(s): December 16, 1996
Published electronically: April 9, 1997
Additional Notes: The first author was supported by NSF grant number DMS-9158092 and by a grant from the Sloan Foundation.
The second author was supported by NSF grant number DMS-9401275.
Communicated by: Alexander Kechris
Article copyright: © Copyright 1997 American Mathematical Society