Left-distributive embedding algebras

Dougherty, Randall¹ ; Jech, Thomas²

¹ Department of Mathematics, Ohio State University, Columbus, OH 43210
² Pennsylvania State University, 215 McAllister Building, University Park, PA 16802

Electronic research announcements of the American Mathematical Society, Tome 03 (1997), pp. 28-37.

Résumé

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.

Détail

DOI : 10.1090/S1079-6762-97-00020-6

Bibliographie

[1] Dehornoy, Patrick Sur la structure des gerbes libres C. R. Acad. Sci. Paris Sér. I Math. 1989 143 148

[2] Dehornoy, Patrick The adjoint representation of left distributive structures Comm. Algebra 1992 1201 1215

[3] Dehornoy, Patrick Braid groups and left distributive operations Trans. Amer. Math. Soc. 1994 115 150

[4] Dehornoy, Patrick From large cardinals to braids via distributive algebra J. Knot Theory Ramifications 1995 33 79

[5] Dougherty, Randall Critical points in an algebra of elementary embeddings Ann. Pure Appl. Logic 1993 211 241

[6] Drápal, Aleš Homomorphisms of primitive left distributive groupoids Comm. Algebra 1994 2579 2592

[7] Drápal, Aleš Persistence of cyclic left distributive algebras J. Pure Appl. Algebra 1995 137 165

[8] Laver, Richard The left distributive law and the freeness of an algebra of elementary embeddings Adv. Math. 1992 209 231

[9] Laver, Richard On the algebra of elementary embeddings of a rank into itself Adv. Math. 1995 334 346

[10] Sieg, Wilfried Fragments of arithmetic Ann. Pure Appl. Logic 1985 33 71

[11] Wehrung, Friedrich Gerbes primitives C. R. Acad. Sci. Paris Sér. I Math. 1991 357 362