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

Nelson Martins-Ferreira 2 , Andrea Montoli 1 , Alex Patchkoria 3 ,  and Manuela Sobral 4
  • 1 Dipartimento di Matematica “Federigo Enriques”, Università degli Studi di Milano, Via Saldini 50, 20133, Milano, Italy
  • 2 Instituto Politécnico de Leiria, ESTG, CDRSP, Leiria, Portugal
  • 3 A.Razmadze Mathematical Institute, Ivane Javakhishvili Tbilisi State University, Tamarashvili Str. 6, Tbilisi, Georgia
  • 4 Departamento de Matemática, Universidade de Coimbra, CMUC, Coimbra, Portugal
Nelson Martins-Ferreira, Andrea Montoli
  • Corresponding author
  • Dipartimento di Matematica “Federigo Enriques”, Università degli Studi di Milano, Via Saldini 50, 20133, Milano, Italy
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, Alex Patchkoria
  • A.Razmadze Mathematical Institute, Ivane Javakhishvili Tbilisi State University, Tamarashvili Str. 6, Tbilisi, 0177, Georgia
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and Manuela Sobral

Abstract

We show that any regular (right) Schreier extension of a monoid M by a monoid A induces an abstract kernel Φ:MEnd(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:MSEnd(A)Inn(A) (resp. Φ:MAut(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 Φ:MSEnd(A)Inn(A) (resp. Φ:MAut(A)Inn(A)), when it is not empty, is in bijection with the second cohomology group of M with coefficients in U(Z(A)).

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