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The index of a subgroup of a group counts the number of cosets of thatsubgroup. A subgroup of finite index often shares structural properties withthe group, and the existence of a subgroup of finite index with some particularproperty can therefore imply useful structural information for the overgroup. Adeveloped theory of cosets in inverse semigroups exists, originally due toSchein: it is defined only for closed inverse subsemigroups, and the structuralcorrespondences between an inverse semigroup and a closed inverse subsemigroupof finite index are weaker than in the group case. Nevertheless, many aspectsof 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 indexformula for chains of subgroups and an analogue of M. Hall's Theorem oncounting subgroups of finite index in finitely generated groups. We then lookin detail at the connection between the following properties of a closedinverse submonoid of an inverse monoid: having finite index; being arecognisable subset; being a rational subset; being finitely generated (as aclosed inverse submonoid). A remarkable result of Margolis and Meakin showsthat these properties are equivalent for closed inverse submonoids of freeinverse monoids.

Inverse subsemigroups of finite index in finitely generated inverse semigroups