Maximal subgroups of amalgams of finite inverse semigroups

We use the description of the Schützenberger automata for amalgams of finite inverse semigroups given by Cherubini et al. (J. Algebra 285:706–725, 2005) to obtain structural results for such amalgams. Schützenberger automata, in the case of amalgams of finite inverse semigroups, are automata with special structure possessing finite subgraphs that contain all the essential information about the automaton. Using this crucial fact, and Bass–Serre theory, we show that the maximal subgroups of an amalgamated free product are either isomorphic to certain subgroups of the original semigroups or can be described as fundamental groups of particular finite graphs of groups built from the maximal subgroups of the original semigroups.