Amalgams of free inverse semigroups

We study inverse semigroup amalgams of the formS * _U T whereS andT are free inverse semigroups andU is an arbitrary finitely generated inverse subsemigroup ofS andT. We make use of recent work of Bennett to show that the word problem is decidable for any such amalgam. This is in contrast to the general situation for semigroup amalgams, where recent work of Birget, Margolis and Meakin shows that the word problem for a semigroup amalgamS * _U T is in general undecidable, even ifS andT have decidable word problem,U is a free semigroup, and the membership problem forU inS andT is decidable. We also obtain a number of results concerning the structure of such amalgams. We obtain conditions for theD-classes of such an amalgam to be finite and we show that the amalgam is combinatorial in such a case. For example every one-relator amalgam of this type has finiteD-classes and is combinatorial. We also obtain information concerning when such an amalgam isE-unitary: for example every one relator amalgam of the formInv<A ∪B :u =v > whereA andB are disjoint andu (resp.v) is a cyclically reduced word overA ∪A ⁻¹ (resp.B ∪B ⁻¹) isE-unitary. Research of all authors supported by a grant from the Italian CNR. The first and third authors’ research was partially supported by MURST. The second author’s research was also partially supported by NSF and the Center for Communication and Information Science of the University of Nebraska at Lincoln.