Adian inverse semigroups

The class of Adian semigroups and Adian groups was first introduced and studied by S. I. Adian in 1966. We introduce the notion of Adian inverse semigroups with the hope that the study of these objects may help in resolving some of the remaining open questions about Adian semigroups and Adian groups. We first prove that Adian inverse semigroups are E-unitary. We then prove that if is a finitely presented Adian inverse semigroup which satisfies the property that the Schützenberger complex of each positive word X+ over the presentation is finite, then the Schützenberger complex of every word over the presentation is finite. As a consequence of this result we are able to solve the word problem for some classes of Adian inverse semigroups, Adian semigroups and Adian groups.