Approaching cosets using Green's relations and Schutzenberger groups

One of the most fundamental concepts in combinatorial group theory is the notion of index. The index of a subgroup is found by counting its right (or left cosets. It may be thought of as providing a way of measuring the difference between a group and a subgroup. In this sense, we can think of finite index subgroups as only differing from their parent group by a finite amount. Many finiteness conditions are known to be preserved under taking finite index subgroups and extensions, including: finite generation / presentability, periodicity, local finiteness, residual finiteness, and having a soluble word problem. Over the past decade or so, several attempts have been made to develop an analogous theory of index for semigroups. In my talk I shall discuss two such approaches (and some recent results relating to them) which arise from two different ways of thinking about what coset should mean for semigroups. The first approach is to think of cosets as being right translates of the substructure under the action of the semigroup on subsets. This approach is restricted in the sense that it only applies usefully to subgroups of semigroups (and not arbitrary subsemigroups). The second approach is a notion of index (which is called the Green index) that arises from a generalised form of Green's relations, where Green's relations are taken relative to a given subsemigroup. This approach has the advantage that it applies to arbitrary subsemigroups. In both cases, theorems exist relating the properties of the semigroup, its subsemigroups, and certain Schutzenberger groups.