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Title: | On the classification of Schreier extensions of monoids with non-abelian kernel |

Authors: | Martins-Ferreira, Nelson Montoli, Andrea Patchkoria, Alex Sobral, Manuela |

Keywords: | Monoid; Schreier extension; obstruction; Eilenberg–Mac Lane cohomology of monoids |

Issue Date: | 2020 |

Publisher: | De Gruyter |

Project: | UID/MAT/00324/2019 |

Serial title, monograph or event: | Forum Mathematicum |

Volume: | 32 |

Issue: | 3 |

Abstract: | We show that any regular (right) Schreier extension of a monoid M by a monoid A induces an abstract kernel \Phi: M \rightarrow\frac{End(A)}{Inn(A)}. If an abstract kernel factors through \frac{SEnd(A)}{Inn(A)}, where SEnd(A) is the monoid of surjective endomorphisms of A, then we associate to it an obstruction, which is an element of the third cohomology group of M with coeffcients in the abelian group U(Z(A)) of invertible elements of the center Z(A) of A, on which M acts via \Phi. An abstract kernel \Phi: M \rightarrow\frac{SEnd(A)}{Inn(A)} (resp. \Phi: M \rightarrow\frac{Aut(A)}{Inn(A)}) is induced by a regular weakly homogeneous (resp. homogeneous) Schreier extension of M by A if and only if its obstruction is zero. We also show that the set of isomorphism classes of regular weakly homogeneous (resp. homogeneous) Schreier extensions inducing a given abstract kernel \Phi: M \rightarrow\frac{SEnd(A)}{Inn(A)} (resp. \Phi: M \rightarrow\frac{Aut(A)}{Inn(A)}), when it is not empty, is in bijection with the second cohomology group of M with coeffcients in U(Z(A)). |

URI: | http://hdl.handle.net/10316/89460 |

DOI: | 10.1515/forum-2019-0164 |

Rights: | embargoedAccess |

Appears in Collections: | I&D CMUC - Artigos em Revistas Internacionais |

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