Let X be a set, X' be a disjoint copy of X and $\bar{X}\wedge\bar{X}=\{(x\wedge y): x,y\in X\cup X'\}$. We look at $\hat{X}=X\cup X'\cup(\bar{X}\wedge \bar{X})$ as a set of letters and consider the free semigroup $\hat{X}^+$ on the set $\hat{X}$. Auinger [1] constructed a model for the bifree locally inverse semigroup on X as a quotient semigroup of $\hat{X}^+$. This result enables us to talk about presentations $\langle X;R\rangle$ of locally inverse semigroups (LI-presentations) where $R\subseteq \hat{X}^+\times\hat{X}^+$. The locally inverse semigroup $LI\langle X;R\rangle$ presented by $\langle X;R\rangle$ is the semigroup $\hat{X}^+/\theta$ where $\theta$ is the least locally inverse semigroup congruence on $\hat{X}^+$ containing $R$. The word problem for $LI\langle X;R\rangle$ is decidable if there exists an algorithm to decide whether or not $(u,v)\in\theta$ for any $u,v\in\hat{X}^+$.

Word problems and other similar questions have been widely studied for other classes of algebras, mainly for groups and for inverse semigroups where graphical methods have been developed to investigate these questions. For inverse semigroups, Stephen [2] introduced the Schützenberger graph of an R-class and used it to address the word problem for inverse semigroup presentations. These graphs reveal themselves to be an important tool and have given a big contribution to the development of combinatorial inverse semigroup theory. However, they work very well for inverse semigroups since the D-classes of these semigroups present a sort of R - L symmetry. For wider classes of semigroups, such as locally inverse semigroups, this symmetry is lost and the Schützenberger graphs lose their utility.

In this seminar we will talk about an ongoing research where we try to generalize Stephen's work to the word problem for LI-presentations. More precisely, we will describe a new type of graph that looks very different from Schützenberger graphs. However, this new type of graph captures the structure of locally inverse semigroups as well as Schützenberger graphs capture the structure of inverse semigroups.

[1] K. Auinger, On the Bifree Locally Inverse Semigroup, J. Algebra 178 (1995), 581-613.

[2] J. B. Stephen, Presentations of inverse semigroups, J. Pure Appl. Algebra 63 (1990), 81-112.