arXiv subjects
Publications and source records attributed to Fabienne Chouraqui.
We establish a correspondence between the invariant subsets of a non-degenerate symmetric set-theoretical solution of the quantum Yang-Baxter equation and the parabolic subgroups of its structure group, equipped with its canonical Garside structure. Moreover, we introduce the notion of a foldable solution, which extends the one of a decomposable solution.
We present an algorithmic approach to the conjugacy problems in monoids and semigroups, using rewriting systems. There is a class of monoids and semigroups that satisfy the condition that the transposi- tion problem and the left and right conjugacy problem are equivalent. The free monoid and the completely simple semigroups belong to this class. We give a solution to the conjugacy problem for monoids and semigroups in this class that are presented by a complete rewriting system that satisfies some additional conditions.
We establish a one-to-one correspondence between structure groups of non-degenerate, involutive and braided "set-theoretical" solutions of the quantum Yang-Baxter equation and Garside groups with a certain presentation. Moreover, we show that the solution is indecomposable if and only if its structure group is a $Δ-$pure Garside group.
In this paper, a connection between rewriting systems and embedding of monoids in groups is found. We show that if a group with a positive presentation has a complete rewriting system $\Re$ that satisfies the condition that each rule in $\Re$ with positive left-hand side has a positive right-hand side, then the monoid presented by the subset of positive rules from $\Re$ embeds in the group. As an example, we give a simple proof that right angled Artin monoids embed in the corresponding right angled Artin groups. This is a special case of the well-known result of Paris \cite{paris} that Artin monoids embed in their groups.
Every tame, prime and alternating knot is equivalent to a tame, prime and alternating knot in regular position, with a common projection. In this work, we show that the Dehn presentation of the knot group of a tame, prime, alternating knot, with a regular and common projection has a finite and complete rewriting system. Although there are rules in the rewriting system with left-hand side a generator and which increase the length of the words we show that the system is terminating.