T. E. Hall proved in 1978 that if [S-1, S-2; U] is amalgam of regular
semigroups in which S-1 boolean AND S-2 = U is a full regular subsemig
roup of S-1 and S-2 (i.e., S-1, S-2, and tr have the same set of idemp
otents), then the amalgam is strongly embeddable in a regular semigrou
p S that contains S-1, S-2, and U as full regular subsemigroups. In th
is case the inductive structure of the amalgamated free produce S-1 (
U) S-2 was studied by Nambooripad and Pastijn in 1989, using Ordman's
results from 1971 on amalgams of groupoids. In the present paper we sh
ow how these results may be combined with techniques from Bass-Serre t
heory to elucidate the structure of the maximal subgroups of S-1 (U)
S-2. This is accomplished by first studying the appropriate analogue o
f the Bass-Serre theory for groupoids and applying this to the study o
f the maximal subgroups of S-1 (U) S-2. The resulting graphs of group
s are arbitrary bipartite graphs of groups, This has several interesti
ng consequences. For example if S, and St are combinatorial, then the
maximal subgroups of S-1 (U) S-2 are free groups. Finite inverse semi
groups may be decomposed in non-trivial ways as amalgams oi inverse se
migroups. (C) 1996 Academic Press, Inc.