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Transfinite braids and left distributive operations

Transfinite braids and left distributive operations Math. Z. 228, 405–433 (1998) c Springer-Verlag 1998 Transfinite braids and left distributive operations Patrick Dehornoy Mathematiques, ´ Universite ´ de Caen, F-14032 Caen, France (e-mail: dehornoy@math.unicaen.fr) Received 23 January 1996; in final form 23 September 1996 Here we investigate a rather natural extension of the usual notion of a braid, namely that obtained by considering two infinite series of strands rather than just one as in the case of Artin’s braid group B . The corresponding group is very large, but it turns out that a certain submonoid EB of this group can be described very simply as a completion of B . This completion is obtained by adding upper bounds to some sequences that are increasing for a canonical linear ordering, or, equivalently, that are Cauchy sequences with respect to the associated topology. The basic study of the monoid EB is the content of the first two sections. The sequel of the paper is devoted to the study of left self-distributive operations on the monoid EB , i.e., of binary operations  that satisfy the algebraic identity x  (y  z)=(x  y)  (x  z): (LD) There is nothing gratuitous in this task. Indeed http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Mathematische Zeitschrift Springer Journals

Transfinite braids and left distributive operations

Dehornoy, Patrick
Mathematische Zeitschrift , Volume 228 (3) – Jul 1, 1998
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References (20)

Publisher
Springer Journals
Copyright
Copyright © 1998 by Springer-Verlag Berlin Heidelberg
Subject
Legacy
ISSN
0025-5874
eISSN
1432-1823
DOI
10.1007/PL00004629
Publisher site
See Article on Publisher Site

Abstract

Math. Z. 228, 405–433 (1998) c Springer-Verlag 1998 Transfinite braids and left distributive operations Patrick Dehornoy Mathematiques, ´ Universite ´ de Caen, F-14032 Caen, France (e-mail: dehornoy@math.unicaen.fr) Received 23 January 1996; in final form 23 September 1996 Here we investigate a rather natural extension of the usual notion of a braid, namely that obtained by considering two infinite series of strands rather than just one as in the case of Artin’s braid group B . The corresponding group is very large, but it turns out that a certain submonoid EB of this group can be described very simply as a completion of B . This completion is obtained by adding upper bounds to some sequences that are increasing for a canonical linear ordering, or, equivalently, that are Cauchy sequences with respect to the associated topology. The basic study of the monoid EB is the content of the first two sections. The sequel of the paper is devoted to the study of left self-distributive operations on the monoid EB , i.e., of binary operations  that satisfy the algebraic identity x  (y  z)=(x  y)  (x  z): (LD) There is nothing gratuitous in this task. Indeed

Journal

Mathematische Zeitschrift – Springer Journals

Published: Jul 1, 1998

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