Cyclic rewriting and conjugacy problems

Cyclic words are equivalence classes of cyclic permutations of ordinary words. When a group is given by a rewriting relation, a rewriting system on cyclic words is induced, which is used to construct algorithms to find minimal length elements of conjugacy classes in the group. These techniques are applied to the universal groups of Stallings pregroups and in particular to free products with amalgamation, HNN-extensions and virtually free groups, to yield simple and intuitive algorithms and proofs of conjugacy criteria.Comment: 37 pages, 1 figure, submitted. Changes to introductio

Towards 3-Dimensional Rewriting Theory

String rewriting systems have proved very useful to study monoids. In good cases, they give finite presentations of monoids, allowing computations on those and their manipulation by a computer. Even better, when the presentation is confluent and terminating, they provide one with a notion of canonical representative of the elements of the presented monoid. Polygraphs are a higher-dimensional generalization of this notion of presentation, from the setting of monoids to the much more general setting of n-categories. One of the main purposes of this article is to give a progressive introduction to the notion of higher-dimensional rewriting system provided by polygraphs, and describe its links with classical rewriting theory, string and term rewriting systems in particular. After introducing the general setting, we will be interested in proving local confluence for polygraphs presenting 2-categories and introduce a framework in which a finite 3-dimensional rewriting system admits a finite number of critical pairs

Coherent Presentations of Monoidal Categories

Presentations of categories are a well-known algebraic tool to provide descriptions of categories by means of generators, for objects and morphisms, and relations on morphisms. We generalize here this notion, in order to consider situations where the objects are considered modulo an equivalence relation, which is described by equational generators. When those form a convergent (abstract) rewriting system on objects, there are three very natural constructions that can be used to define the category which is described by the presentation: one consists in turning equational generators into identities (i.e. considering a quotient category), one consists in formally adding inverses to equational generators (i.e. localizing the category), and one consists in restricting to objects which are normal forms. We show that, under suitable coherence conditions on the presentation, the three constructions coincide, thus generalizing celebrated results on presentations of groups, and we extend those conditions to presentations of monoidal categories

Globalization of Confluent Partial Actions on Topological and Metric Spaces

We generalize Exel's notion of partial group action to monoids. For partial monoid actions that can be defined by means of suitably well-behaved systems of generators and relations, we employ classical rewriting theory in order to describe the universal induced global action on an extended set. This universal action can be lifted to the setting of topological spaces and continuous maps, as well as to that of metric spaces and non-expansive maps. Well-known constructions such as Shimrat's homogeneous extension are special cases of this construction. We investigate various properties of the arising spaces in relation to the original space; in particular, we prove embedding theorems and preservation properties concerning separation axioms and dimension. These results imply that every normal (metric) space can be embedded into a normal (metrically) ultrahomogeneous space of the same dimension and cardinality.Comment: New presentation of material on rewritin

Coherent presentation for the hypoplactic monoid of rank n

In this thesis, we construct a coherent presentation for the hypoplactic monoid of rank n and characterize the confluence diagrams associated with it, then we use the theory of quasi-Kashiwara operators and quasi-crystal graphs to prove that all confluence diagrams can be obtained from those diagrams whose vertices are highest-weight words. To do so, we first give a complete rewriting system for the hypoplactic monoid of rank n, then, using an extension of the Knuth–Bendix completion procedure called the homotopical completion procedure, we compute the previously mentioned coherent presentation, which, from a viewpoint of Monoidal Category Theory, gives us a family of generators of the relations amongst the relations. These coherent presentations are used for representations of monoids and are particularly useful to describe actions of monoids on categories. The theoretical background is given without proof, since the main purpose of this thesis is to present new results