### School of Mathematics and Statistics

### Michael Hoffmann

University of Leicester

### Automatic and Biautomatic Semigroups

**Friday 13th August, 12-1pm, Carslaw 375. **
This talk describes some joint work with Rick Thomas from Leicester.

There has been a great deal of interest in recent years in the
theory of automatic and biautomatic groups. As with the notion of
automaticity, the idea of biautomaticity can be generalized from
groups to semigroups. As with automaticity, the definition extends
naturally, but new proof techniques are often required in the
semigroup environment.

A major open question in group theory is whether or not an automatic
group is necessarily biautomatic. In groups the notions of
left-automatic and right-automatic coincide (one simply describes
this as "automatic"), but we point out that this is not the case in
semigroups (where "automatic" is taken to mean "right-automatic").
As a consequence, there are automatic semigroups which are not
biautomatic. In fact, there are examples of semigroups that are both
left- and right-automatic but not biautomatic, and we will briefly
describe such an example. We will also indicate how all this sheds
some light on the theory of asynchronous automatic groups.

The fellow traveller property can be used to characterise automatic
groups. It has been shown very useful for proving results for
automatic groups. For automatic semigroups the fellow traveller
property is necessary but not sufficient. For semigroups with
bounded indegree we will give another geometric condition for the
Cayley graph of a semigroup S which characterises if S is automatic.
This condition is not in a 'nice' form and we are still working on
it to bring it to a 'nice' and easier to use shape.

Results on automatic semigroups with subsemigroups of finite
Rees-Index and on the presentation of automatic semigroups show the
wide range of this interesting area.