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 and one generator, satisfying the left distributive law ; 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 , and then take a limit to get an infinite left-distributive algebra on one generator. Results of Laver and Steel assuming a strong large cardinal axiom imply that is free; it is open whether the freeness of 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 is unprovable in primitive recursive arithmetic.

**1.**Patrick Dehornoy,*Sur la structure des gerbes libres*, C. R. Acad. Sci. Paris Sér. I Math.**309**(1989), no. 3, 143–148 (French, with English summary). MR**1005627****2.**Patrick Dehornoy,*The adjoint representation of left distributive structures*, Comm. Algebra**20**(1992), no. 4, 1201–1215. MR**1154409**, https://doi.org/10.1080/00927879208824398**3.**Patrick Dehornoy,*Braid groups and left distributive operations*, Trans. Amer. Math. Soc.**345**(1994), no. 1, 115–150. MR**1214782**, https://doi.org/10.1090/S0002-9947-1994-1214782-4**4.**Patrick Dehornoy,*From large cardinals to braids via distributive algebra*, J. Knot Theory Ramifications**4**(1995), no. 1, 33–79. MR**1321290**, https://doi.org/10.1142/S0218216595000041**5.**Randall Dougherty,*Critical points in an algebra of elementary embeddings*, Ann. Pure Appl. Logic**65**(1993), no. 3, 211–241. MR**1263319**, https://doi.org/10.1016/0168-0072(93)90012-3**6.**-,*Critical points in an algebra of elementary embeddings, II*, Logic: from Foundations to Applications (W. Hodges*et al.*, eds.), Clarendon, Oxford, 1996, pp. 103-136. CMP**97:06****7.**R. Dougherty and T. Jech,*Finite left-distributive algebras and embedding algebras*, Adv. Math. (to appear).**8.**Aleš Drápal,*Homomorphisms of primitive left distributive groupoids*, Comm. Algebra**22**(1994), no. 7, 2579–2592. MR**1271624**, https://doi.org/10.1080/00927879408824978**9.**Aleš Drápal,*Persistence of cyclic left distributive algebras*, J. Pure Appl. Algebra**105**(1995), no. 2, 137–165. MR**1365873**, https://doi.org/10.1016/0022-4049(94)00142-1**10.**Richard Laver,*The left distributive law and the freeness of an algebra of elementary embeddings*, Adv. Math.**91**(1992), no. 2, 209–231. MR**1149623**, https://doi.org/10.1016/0001-8708(92)90016-E**11.**Richard Laver,*On the algebra of elementary embeddings of a rank into itself*, Adv. Math.**110**(1995), no. 2, 334–346. MR**1317621**, https://doi.org/10.1006/aima.1995.1014**12.**Wilfried Sieg,*Fragments of arithmetic*, Ann. Pure Appl. Logic**28**(1985), no. 1, 33–71. MR**776285**, https://doi.org/10.1016/0168-0072(85)90030-2**13.**Friedrich Wehrung,*Gerbes primitives*, C. R. Acad. Sci. Paris Sér. I Math.**313**(1991), no. 6, 357–362 (French, with English summary). MR**1126413**

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