|Ph.D Student||Fabienne Chouraqui|
|Subject||Decision Problems in Tableau-Groups and Tableau-Semigroups|
|Department||Department of Mathematics||Supervisor||Professor Juhasz Arie|
|Full Thesis text|
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* , u1=u, u2,.., uk and we apply on each element ui rules from ℜ . We say that u is cyclically irreducible if u and all its cyclic conjugates are irreducible modulo ℜ. If for some i, 1 ≤ i ≤ n, ui reduces to v, then we say that u cyclically reduces to v. A question that arises naturally is when u and all its cyclic conjugates cyclically reduce to the same cyclically irreducible element (up to cyclic conjugation in Σ*), denoted by ρ(u). We find, given a word u in Σ*, a criteria that ensures the existence of a unique cyclically irreducible element ρ(u) . Moreover, the answer to this question gives a partial solution to the conjugacy problems presented above in the following way: if u and v are transposed, then ρ(u) and ρ(v) are cyclic conjugates in Σ* and this implies in turn that u and v are left and right conjugates .