On the classification of Schreier extensions of monoids with non-abelian kernel
Autor(a) principal: | |
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Data de Publicação: | 2020 |
Outros Autores: | , , |
Tipo de documento: | Artigo |
Idioma: | eng |
Título da fonte: | Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
Texto Completo: | http://hdl.handle.net/10316/89460 https://doi.org/10.1515/forum-2019-0164 |
Resumo: | 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)). |
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Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
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7160 |
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On the classification of Schreier extensions of monoids with non-abelian kernelMonoid; Schreier extension; obstruction; Eilenberg–Mac Lane cohomology of monoidsWe 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)).De Gruyter2020info:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/articlehttp://hdl.handle.net/10316/89460http://hdl.handle.net/10316/89460https://doi.org/10.1515/forum-2019-0164enghttps://www.degruyter.com/view/journals/form/32/3/article-p607.xmlMartins-Ferreira, NelsonMontoli, AndreaPatchkoria, AlexSobral, Manuelainfo:eu-repo/semantics/openAccessreponame:Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)instname:Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informaçãoinstacron:RCAAP2022-05-25T06:33:50Zoai:estudogeral.uc.pt:10316/89460Portal AgregadorONGhttps://www.rcaap.pt/oai/openaireopendoar:71602024-03-19T21:09:46.035480Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) - Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informaçãofalse |
dc.title.none.fl_str_mv |
On the classification of Schreier extensions of monoids with non-abelian kernel |
title |
On the classification of Schreier extensions of monoids with non-abelian kernel |
spellingShingle |
On the classification of Schreier extensions of monoids with non-abelian kernel Martins-Ferreira, Nelson Monoid; Schreier extension; obstruction; Eilenberg–Mac Lane cohomology of monoids |
title_short |
On the classification of Schreier extensions of monoids with non-abelian kernel |
title_full |
On the classification of Schreier extensions of monoids with non-abelian kernel |
title_fullStr |
On the classification of Schreier extensions of monoids with non-abelian kernel |
title_full_unstemmed |
On the classification of Schreier extensions of monoids with non-abelian kernel |
title_sort |
On the classification of Schreier extensions of monoids with non-abelian kernel |
author |
Martins-Ferreira, Nelson |
author_facet |
Martins-Ferreira, Nelson Montoli, Andrea Patchkoria, Alex Sobral, Manuela |
author_role |
author |
author2 |
Montoli, Andrea Patchkoria, Alex Sobral, Manuela |
author2_role |
author author author |
dc.contributor.author.fl_str_mv |
Martins-Ferreira, Nelson Montoli, Andrea Patchkoria, Alex Sobral, Manuela |
dc.subject.por.fl_str_mv |
Monoid; Schreier extension; obstruction; Eilenberg–Mac Lane cohomology of monoids |
topic |
Monoid; Schreier extension; obstruction; Eilenberg–Mac Lane cohomology of monoids |
description |
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)). |
publishDate |
2020 |
dc.date.none.fl_str_mv |
2020 |
dc.type.status.fl_str_mv |
info:eu-repo/semantics/publishedVersion |
dc.type.driver.fl_str_mv |
info:eu-repo/semantics/article |
format |
article |
status_str |
publishedVersion |
dc.identifier.uri.fl_str_mv |
http://hdl.handle.net/10316/89460 http://hdl.handle.net/10316/89460 https://doi.org/10.1515/forum-2019-0164 |
url |
http://hdl.handle.net/10316/89460 https://doi.org/10.1515/forum-2019-0164 |
dc.language.iso.fl_str_mv |
eng |
language |
eng |
dc.relation.none.fl_str_mv |
https://www.degruyter.com/view/journals/form/32/3/article-p607.xml |
dc.rights.driver.fl_str_mv |
info:eu-repo/semantics/openAccess |
eu_rights_str_mv |
openAccess |
dc.publisher.none.fl_str_mv |
De Gruyter |
publisher.none.fl_str_mv |
De Gruyter |
dc.source.none.fl_str_mv |
reponame:Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) instname:Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação instacron:RCAAP |
instname_str |
Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação |
instacron_str |
RCAAP |
institution |
RCAAP |
reponame_str |
Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
collection |
Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
repository.name.fl_str_mv |
Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) - Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação |
repository.mail.fl_str_mv |
|
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1799133992899837952 |