155 research outputs found

    Monoids of O-type, subword reversing, and ordered groups

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    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

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    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

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    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

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    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

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    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

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    The braids of B_∞B\_\infty can be equipped with a selfdistributive operation ▹\mathbin{\triangleright} 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

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    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

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    We develop a new approach to the linear ordering of the braid group B_nB\_n, based on investigating its restriction to the set \Div(\Delta\_n^d) of all divisors of Δ_nd\Delta\_n^d in the monoid B_∞+B\_\infty^+, i.e., to positive nn-braids whose normal form has length at most dd. 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 B_3B\_3, and a new proof of the result that its restriction to B_3+B\_3^+ is a well-ordering of ordinal type ωω\omega^\omega

    Using groups for investigating rewrite systems

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    We describe several technical tools that prove to be efficient for investigating the rewrite systems associated with a family of algebraic laws, and might be useful for more general rewrite systems. These tools consist in introducing a monoid of partial operators, listing the monoid relations expressing the possible local confluence of the rewrite system, then introducing the group presented by these relations, and finally replacing the initial rewrite system with a internal process entirely sitting in the latter group. When the approach can be completed, one typically obtains a practical method for constructing algebras satisfying prescribed laws and for solving the associated word problem
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