BASS-SERRE THEORY FOR GROUPOIDS AND THE STRUCTURE OF FULL REGULAR SEMIGROUP AMALGAMS

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.