155 research outputs found
Monoids of O-type, subword reversing, and ordered groups
We describe a simple scheme for constructing finitely generated monoids in
which left-divisibility is a linear ordering and for practically investigating
these monoids. The approach is based on subword reversing, a general method of
combinatorial group theory, and connected with Garside theory, here in a
non-Noetherian context. As an application we describe several families of
ordered groups whose space of left-invariant orderings has an isolated point,
including torus knot groups and some of their amalgamated products.Comment: updated version with new result
The group of fractions of a torsion free lcm monoid is torsion free
We improve and shorten the argument given in(Journal of Algebra, vol.~210
(1998) pp~291--297). Inparticular, the fact that Artin braid groups are torsion
free now follows from Garside\'s results almost immediately
Some aspects of the SD-world
We survey a few of the many results now known about the self-distributivity
law and selfdistributive structures, with a special emphasis on the associated
word problems and the algorithms solving them in good cases
On the rotation distance between binary trees
We develop combinatorial methods for computing the rotation distance between
binary trees, i.e., equivalently, the flip distance between triangulations of a
polygon. As an application, we prove that, for each n, there exist size n trees
at distance 2n - O(sqrt(n))
Multifraction reduction I: The 3-Ore case and Artin-Tits groups of type FC
We describe a new approach to the Word Problem for Artin-Tits groups and,
more generally, for the enveloping group U(M) of a monoid M in which any two
elements admit a greatest common divisor. The method relies on a rewrite system
R(M) that extends free reduction for free groups. Here we show that, if M
satisfies what we call the 3-Ore condition about common multiples, what
corresponds to type FC in the case of Artin-Tits monoids, then the system R(M)
is convergent. Under this assumption, we obtain a unique representation result
for the elements of U(M), extending Ore's theorem for groups of fractions and
leading to a solution of the Word Problem of a new type. We also show that
there exist universal shapes for the van Kampen diagrams of the words
representing 1.Comment: 29 pages ; v2 : cross-references updated ; v3 : typos corrected;
final version due to appear in Journal of Combinatorial Algebr
The Braid Shelf
The braids of can be equipped with a selfdistributive operation
enjoying a number of deep properties. This text is a
survey of known properties and open questions involving this structure, its
quotients, and its extensions
Using shifted conjugacy in braid-based cryptography
Conjugacy is not the only possible primitive for designing braid-based
protocols. To illustrate this principle, we describe a Fiat--Shamir-style
authentication protocol that be can be implemented using any binary operation
that satisfies the left self-distributive law. Conjugation is an example of
such an operation, but there are other examples, in particular the shifted
conjugation on Artin's braid group B\_oo, and the finite Laver tables. In both
cases, the underlying structures have a high combinatorial complexity, and they
lead to difficult problems
Still another approach to the braid ordering
We develop a new approach to the linear ordering of the braid group ,
based on investigating its restriction to the set \Div(\Delta\_n^d) of all
divisors of in the monoid , i.e., to positive
-braids whose normal form has length at most . In the general case, we
compute several numerical parameters attached with the finite orders
(\Div(\Delta\_n^d), <). In the case of 3 strands, we moreover give a complete
description of the increasing enumeration of (\Div(\Delta\_3^d), <). We
deduce a new and specially direct construction of the ordering on , and a
new proof of the result that its restriction to is a well-ordering of
ordinal type
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