On the decidability of the word problem for amalgamated free products of inverse semigroups.

A triple [S,T;U] where S and T are inverse semigroups and U is the intersection of S and T is called an amalgam of inverse semigroups. The amalgamated free product associated to the amalgam [S,T;U] is defined by the usual universal diagram and it is well-known that each amalgams of inverse semigroups embeds in its associated aalgamated free product. In general amalgamated free products of inverse semigroups have undecidable word problem even when S, T, U satisfy nice conditions. Here we consider amalgams of inverse semigroups where S and T have decidable word problem, U has decidable membership problem and we give some sufficient conditions for the associated amalgamated free product having decidable word problem