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

On the classification of Schreier extensions of monoids with non-abelian kernel AbstractWe 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){\Phi\colon M\to\frac{\operatorname{End}(A)}{\operatorname{Inn}(A)}}.If an abstract kernel factors through SEnd⁡(A)Inn⁡(A){\frac{\operatorname{SEnd}(A)}{\operatorname{Inn}(A)}}, where SEnd⁡(A){\operatorname{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)){U(Z(A))}of invertible elements of the center Z⁢(A){Z(A)}of A, on which M acts via Φ.An abstract kernel Φ:M→SEnd⁡(A)Inn⁡(A){\Phi\colon M\to\frac{\operatorname{SEnd}(A)}{\operatorname{Inn}(A)}}(resp. Φ:M→Aut⁡(A)Inn⁡(A){\Phi\colon M\to\frac{\operatorname{Aut}(A)}{\operatorname{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:M→SEnd⁡(A)Inn⁡(A){\Phi\colon M\to\frac{\operatorname{SEnd}(A)}{\operatorname{Inn}(A)}}(resp. Φ:M→Aut⁡(A)Inn⁡(A){\Phi\colon M\to\frac{\operatorname{Aut}(A)}{\operatorname{Inn}(A)}}), when it is not empty, is in bijection with the second cohomology group of M with coefficients in U⁢(Z⁢(A)){U(Z(A))}. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Forum Mathematicum de Gruyter

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

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Publisher
de Gruyter
Copyright
© 2020 Walter de Gruyter GmbH, Berlin/Boston
ISSN
0933-7741
eISSN
1435-5337
DOI
10.1515/forum-2019-0164
Publisher site
See Article on Publisher Site

Abstract

AbstractWe 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){\Phi\colon M\to\frac{\operatorname{End}(A)}{\operatorname{Inn}(A)}}.If an abstract kernel factors through SEnd⁡(A)Inn⁡(A){\frac{\operatorname{SEnd}(A)}{\operatorname{Inn}(A)}}, where SEnd⁡(A){\operatorname{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)){U(Z(A))}of invertible elements of the center Z⁢(A){Z(A)}of A, on which M acts via Φ.An abstract kernel Φ:M→SEnd⁡(A)Inn⁡(A){\Phi\colon M\to\frac{\operatorname{SEnd}(A)}{\operatorname{Inn}(A)}}(resp. Φ:M→Aut⁡(A)Inn⁡(A){\Phi\colon M\to\frac{\operatorname{Aut}(A)}{\operatorname{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:M→SEnd⁡(A)Inn⁡(A){\Phi\colon M\to\frac{\operatorname{SEnd}(A)}{\operatorname{Inn}(A)}}(resp. Φ:M→Aut⁡(A)Inn⁡(A){\Phi\colon M\to\frac{\operatorname{Aut}(A)}{\operatorname{Inn}(A)}}), when it is not empty, is in bijection with the second cohomology group of M with coefficients in U⁢(Z⁢(A)){U(Z(A))}.

Journal

Forum Mathematicumde Gruyter

Published: May 1, 2020

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