Inverse subsemigroups of finite index in finitely generated inverse semigroups

The index of a subgroup of a group counts the number of cosets of that subgroup. A subgroup of finite index often shares structural properties with the group, and the existence of a subgroup of finite index with some particular property can therefore imply useful structural information for the overgroup. A developed theory of cosets in inverse semigroups exists, originally due to Schein: it is defined only for closed inverse subsemigroups, and the structural correspondences between an inverse semigroup and a closed inverse subsemigroup of finite index are weaker than in the group case. Nevertheless, many aspects of this theory are of interest, and some of them are addressed in this paper. We study the basic theory of cosets in inverse semigroups, including an index formula for chains of subgroups and an analogue of M. Hall's Theorem on counting subgroups of finite index in finitely generated groups. We then look in detail at the connection between the following properties of a closed inverse submonoid of an inverse monoid: having finite index; being a recognisable subset; being a rational subset; being finitely generated (as a closed inverse submonoid). A remarkable result of Margolis and Meakin shows that these properties are equivalent for closed inverse submonoids of free inverse monoids.