Algebra Seminar

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.