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Left-distributive embedding algebras
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## Left-distributive embedding algebras

### Randall Dougherty and Thomas Jech

**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 {\em 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.

*Copyright 1997 American Mathematical Society*

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#### Article Info

- ERA Amer. Math. Soc.
**03** (1997), pp. 28-37
- Publisher Identifier: S 1079-6762(97)00020-6
- 1991
*Mathematics Subject Classification*. Primary 20N02; Secondary 03E55, 08B20
*Key words and phrases*. Left-distributive algebras, elementary embeddings, critical
points, large cardinals, primitive recursive arithmetic
- Received by the editors December 16, 1996
- Posted on April 9, 1997
- Communicated by Alexander Kechris
- Comments (When Available)

**Randall Dougherty**

Department of Mathematics, Ohio State University,
Columbus, OH 43210

*E-mail address:* `rld@math.ohio-state.edu `

**Thomas Jech**

Pennsylvania State University, 215 McAllister Building,
University Park, PA 16802

*E-mail address:* `jech@math.psu.edu `

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.

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