Amalgams of free inverse semigroups

We study inverse semigroup amalgams of the form S *(u) T where S and T are free inverse semigroups and U is an arbitrary finitely generated inverse subsemigroup of S and T. 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 amalgam S *(u) T is in general undecidable, even if S and T have decidable word problem, U is a free semigroup, and the membership problem for U in S and T is decidable. We also obtain a number of results concerning the structure of such amalgams. We obtain conditions for the D-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 finite D-classes and is combinatorial. We also obtain information concerning when such an amalgam is E-unitary: for example every one relator amalgam of the form Inv < A boolean OR B : u = nu > where A and B are disjoint and u (resp. nu) is a cyclically reduced word over A boolean OR A(-1) (resp. B boolean OR B-1) is E-unitary.