Ph.D Thesis | |

Ph.D Student | Chouraqui Fabienne |
---|---|

Subject | Decision Problems in Tableau-Groups and Tableau-Semigroups |

Department | Department of Mathematics |

Supervisor | ASSOCIATE PROFESSOR Arie Juhasz |

This work is
about decision problems in some class of groups and monoids. We define a *tableau-group
(semigroup)* to be a finitely presented group (semigroup) such that all the
defining relations have positive words of length two on both sides of the
equality .
There is a lot of examples of such groups: Knot and link groups with the
Wirtinger presentation, right-angled Artin groups, LOG groups (Labeled Oriented
Graph), structure groups of set-theoretical solutions of the quantum
Yang-Baxter equation and others . The class of tableau-groups or semigroups is closed under free,
direct and graph product. Our research considers some subclasses of
tableau-groups and monoids and we study their algorithmic and algebraic
properties .

We establish a one-to-one
correspondence between the structure groups of non degenerate, involutive and braided
set-theoretical solutions of the quantum Yang-Baxter equation and a class of
Garside tableau groups that satisfies certain conditions . Our approach is combinatorial and we use
the tools developed by Dehornoy in the theory of Garside groups . We study the
conjugacy problem in semigroups and monoids. There are several notions: the
transposition problem, the left (right) conjugacy problem and the left and
right conjugacy problem . The main idea in our proof can be described as follows .
Let *M* be a finitely presented monoid generated by Σ and let ℜ be a complete
rewriting system for *M* . Let *u* be a word in Σ^{*},
the free monoid generated by Σ.
We consider *u* and all its cyclic conjugates in Σ^{*} , *u _{1}=u,
u_{2,..,} u_{k }*and we apply on each element