On the classification of Schreier extensions of monoids with non-abelian kernel (Preprint)

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Type: Preprint
National /International: International
Title: On the classification of Schreier extensions of monoids with non-abelian kernel
Publication Date: 2019-06-25
Authors: - Nelson Martins-Ferreira
- Andrea Montoli
- Alex Patchkoria
- Manuela Sobral
Abstract: We show that any regular (right) Schreier extension of a monoid M by a monoid A induces an abstract kernel Φ: M → End(A) / Inn(A) . If an abstract kernel factors through 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 coefficients in the abelian group U(Z(A)) of invertible elements of the center Z(A) of A, on which M acts via Φ. An abstract kernel Φ: M → SEnd(A) / Inn(A) (resp. Φ: M → 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 isomorphic classes of regular weakly homogeneous (resp. homogeneous) Schreier extensions inducing a given abstract kernel Φ: M → SEnd(A) / Inn(A) (resp. Φ: M → Aut(A) / Inn(A) ), when it is not empty, is in bijection with the second cohomology group of M with coefficients in U(Z(A)).
Institution: DMUC 19-23
Online version: http://www.mat.uc.pt...prints/eng_2019.html
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