Bass–Serre Theory for Groupoids and the Structure of Full Regular Semigroup Amalgams Original Research Article

T. E. Hall proved in 1978 that if [S1, S2; U] is an amalgam of regular semigroups in whichS1∩S2=Uis a full regular subsemigroup ofS1andS2(i.e.,S1,S2, andUhave the same set of idempotents), then the amalgam is strongly embeddable in a regular semigroupSthat containsS1,S2, andUas full regular subsemigroups. In this case the inductive structure of the amalgamated free produceS1*US2was studied by Nambooripad and Pastijn in 1989, using Ordmanʹs results from 1971 on amalgams of groupoids. In the present paper we show how these results may be combined with techniques from Bass–Serre theory to elucidate the structure of the maximal subgroups ofS1*US2. This is accomplished by first studying the appropriate analogue of the Bass–Serre theory for groupoids and applying this to the study of the maximal subgroups ofS1*US2. The resulting graphs of groups are arbitrary bipartite graphs of groups. This has several interesting consequences. For example ifS1andS2are combinatorial, then the maximal subgroups ofS1*US2are free groups. Finite inverse semigroups may be decomposed in non-trivial ways as amalgams of inverse semigroups.