Searcharxiv⌕ Search

arXiv subjects

Patrick Dehornoy

Publications and source records attributed to Patrick Dehornoy.

At least 19 recordsLinked to original sources
arXiv2018

A cancellativity criterion for presented monoids

We establish a new, fairly general cancellativity criterion for a presented monoid that properly extends the previously known related criteria. It is based on a new version of the word transformation called factor reversing, and its specificity is to avoid any restriction on the number of relations in the presentation. As an application, we deduce the cancellativity of some natural extension of Artin's braid monoid in which crossings are colored.

math.GR↗
arXiv2018

Garside combinatorics for Thompson's monoid $F^+$ and a hybrid with the braid monoid $B\_\infty^+$

On the model of simple braids, defined to be the left divisors of Garside's elements $Δ\_n$ in the monoid $B\_\infty^+$ , we investigate simple elements in Thompson's monoid $F^+$ and in a larger monoid $H^+$ that is a hybrid of $B\_\infty^+$ and $F^+$ : in both cases, we count how many simple elements left divide the right lcm of the first n -- 1 atoms, and characterize their normal forms in terms of forbidden factors. In the case of $H^+$, a generalized Pascal triangle appears.

math.GR↗
arXiv2018

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.

math.GR↗
arXiv2018

The Braid Shelf

The braids of $B\_\infty$ 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.

math.GR↗
arXiv2017

Multifraction reduction II: Conjectures for Artin-Tits groups

Multifraction reduction is a new approach to the word problem for Artin-Tits groups and, more generally, for the enveloping group of a monoid in which any two elements admit a greatest common divisor. This approach is based on a rewrite system ("reduction") that extends free group reduction. In this paper, we show that assuming that reduction satisfies a weak form of convergence called semi-convergence is sufficient for solving the word problem for the enveloping group, and we connect semi-convergence with other conditions involving reduction. We conjecture that these properties are valid for all Artin-Tits monoids, and provide partial results and numerical evidence supporting such conjectures.

math.GR↗
arXiv2017

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.

math.GR↗
arXiv2017

Multifraction reduction III: The case of interval monoids

We investigate gcd-monoids, which are cancellative monoids in which any two elements admit a left and a right gcd, and the associated reduction of multifractions (arXiv:1606.08991 and 1606.08995), a general approach to the word problem for the enveloping group. Here we consider the particular case of interval monoids associated with finite posets. In this way, we construct gcd-monoids, in which reduction of multifractions has prescribed properties not yet known to be compatible: semi-convergence of reduction without convergence, semi-convergence up to some level but not beyond, non-embeddability into the enveloping group (a strong negation of semi-convergence).

math.GR↗
arXiv2017

Multifraction reduction IV: Padding and Artin-Tits groups of sufficiently large type

We investigate the padded version of reduction, an extension of multifraction reduction as defined in arXiv:1606.08991, and connect it both with ordinary reduction and with the so-called Property $\mathrm{H}$. As an application, we show that all Artin-Tits groups of sufficiently large type satisfy some weakening Conjecture $\mathrm{A^{padded}}$ of Conjecture $\mathrm{A}$, thus showing that the reduction approach is relevant for these groups.

math.GR↗
arXiv2016

Quadratic normalisation in monoids

In the general context of presentations of monoids, we study normalisation processes that are determined by their restriction to length-two words. Garside's greedy normal forms and quadratic convergent rewriting systems, in particular those associated with the plactic monoids, are typical examples. Having introduced a parameter, called the class and measuring the complexity of the normalisation of length-three words, we analyse the normalisation of longer words and describe a number of possible behaviours. We fully axiomatise normalisations of class (4, 3), show the convergence of the associated rewriting systems, and characterise those deriving from a Garside family.

math.GR↗
arXiv2015

Garside and quadratic normalisation: a survey

Starting from the seminal example of the greedy normal norm in braid monoids, we analyse the mechanism of the normal form in a Garside monoid and explain how it extends to the more general framework of Garside families. Extending the viewpoint even more, we then consider general quadratic normalisation procedures and characterise Garside normalisation among them.

math.GR↗
arXiv2014

Addenda to "Foundations of Garside Theory"

This text consists of additions to the book "Foundations of Garside Theory", EMS Tracts in Mathematics, vol. 22 (2015) -- see introduction and table of contents in arXiv:1309.0796 -- namely skipped proofs and solutions to selected exercises.

math.GR↗
arXiv2014

Garside families in Artin-Tits monoids and low elements in Coxeter groups

We show that every finitely generated Artin-Tits group admits a finite Garside family, by introducing the notion of a low element in a Coxeter group and proving that the family of all low elements in a Coxeter system (W, S) with S finite includes S and is finite and closed under suffix and join with respect to the right weak order.

math.GR↗
arXiv2014

Dual Garside structure of braids and free cumulants of products

We count the n-strand braids whose normal decomposition has length at most two in the dual braid monoid B_n+* by reducing the question to a computation of free cumulants for a product of independent variables, for which we establish a general formula.

math.CO↗
arXiv2014

Set-theoretic solutions of the Yang-Baxter equation, RC-calculus, and Garside germs

Building on a result by W. Rump, we show how to exploit the right-cyclic law (x.y).(x.z) = (y.x).(y.z) in order to investigate the structure groups and monoids attached with (involutive nondegenerate) set-theoretic solutions of the Yang-Baxter equation. We develop a sort of right-cyclic calculus, and use it to obtain short proofs for the existence both of the Garside structure and of the I-structure of such groups. We describe finite quotients that exactly play for the considered groups the role that Coxeter groups play for Artin-Tits groups.

math.GR↗
arXiv2014

Two- and three-cocycles for Laver tables

We determine all 2- and 3-cocycles for Laver tables, an infinite sequence of finite structures obeying the left-selfdistributivity law; in particular, we describe simple explicit bases. This provides a number of new positive braid invariants and paves the way for further potential topological applications. Incidentally, we establish and study a partial ordering on Laver tables given by the right-divisibility relation.

math.KT↗
arXiv2014

Laver's results and low-dimensional topology

In connection with his interest in selfdistributive algebra, Richard Laver established two deep results with potential applications in low-dimensional topology, namely the existence of what is now known as the Laver tables and the well-foundedness of the standard ordering of positive braids. Here we present these results and discuss the way they could be used in topological applications.

math.GT↗
arXiv2013

Foundations of Garside Theory

This text consists of the introduction, table of contents, and bibliography of a long manuscript (703 pages) that is currently submitted for publication. This manuscript develops an extension of Garside's approach to braid groups and provides a unified treatment for the various algebraic structures that appear in this context. The complete text can be found at http://www.math.unicaen.fr/~garside/Garside.pdf.

math.GR↗