Remote Access Electronic Research Announcements

Electronic Research Announcements

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
DOI: https://doi.org/10.1090/S1079-6762-97-00020-6
Published electronically: April 9, 1997
MathSciNet review: 1445632
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Electronic Research Announcements of the American Mathematical Society with MSC (1991): 20N02, 03E55, 08B20

Retrieve articles in all journals with MSC (1991): 20N02, 03E55, 08B20


Additional Information

Randall Dougherty
Affiliation: Department of Mathematics, Ohio State University, Columbus, OH 43210
Email: rld@math.ohio-state.edu

Thomas Jech
Affiliation: Pennsylvania State University, 215 McAllister Building, University Park, PA 16802
Email: jech@math.psu.edu

DOI: https://doi.org/10.1090/S1079-6762-97-00020-6
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