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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)).
DOI: 10.1515/forum-2019-0164
Rights: embargoedAccess
Appears in Collections:I&D CMUC - Artigos em Revistas Internacionais

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