## Abstract

T. E. Hall proved in 1978 that if [S_{1}, S_{2}; U] is an amalgam of regular semigroups in which S_{1} ∩ S_{2} = U is a full regular subsemigroup of S_{1} and S_{2} (i.e., S_{1}, S_{2}, and U have the same set of idempotents), then the amalgam is strongly embeddable in a regular semigroup S that contains S_{1}, S_{2}, and U as full regular subsemigroups. In this 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 show how these results may be combined with techniques from Bass-Serre theory to elucidate the structure of the maximal subgroups of S_{1} *_{U} S_{2}. 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 of S_{1} *_{U} S_{2}. The resulting graphs of groups are arbitrary bipartite graphs of groups. This has several interesting consequences. For example if S_{1} and S_{2} are combinatorial, then the maximal subgroups of S_{1} *_{U} S_{2} are free groups. Finite inverse semigroups may be decomposed in non-trivial ways as amalgams of inverse semigroups.

Original language | English |
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Pages (from-to) | 38-54 |

Number of pages | 17 |

Journal | Journal of Algebra |

Volume | 183 |

Issue number | 1 |

DOIs | |

State | Published - 1 Jul 1996 |

Externally published | Yes |

### Bibliographical note

Funding Information:* Research supported by the NSF and the Center for Communication and Information Science of the University of Nebraska at Lincoln. ²E-mail: [email protected]. ³E-mail: [email protected].

### Funding

* Research supported by the NSF and the Center for Communication and Information Science of the University of Nebraska at Lincoln. ²E-mail: [email protected]. ³E-mail: [email protected].

Funders | Funder number |
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Center for Communication and Information Science of the University of Nebraska | |

National Science Foundation |