## University of Sydney Algebra Seminar

# David Easdown (University of Sydney)

## Friday 16 December, 12:05-12:55pm, Carslaw 375

### Presentations of maximal subgroups of semigroups generated by idempotents subject to relations that come from the underlying biordered set

We discuss a geometric construction, using fundamental groups of graphs, to produce presentations of maximal subgroups of semigroups generated by elements of a biordered set, subject to relations associated with basic products. We relate this to recent combinatorial results of Gray and Ruskuc (2011) that show, in particular, that the semigroup presentation has unsolvable word problem. We introduce the original notion of biorder due to Nambooripad (1975) and Easdown's arrow diagrams (1985), illustrate the notion of singular squares in a biordered set, and how they lead to geometric interpretations that explain interesting phenomena involving recent break-through examples of Dolinka (2011), Brittenham, Margolis and Meakin (2009). This is joint work with Brett McElwee.