On the classification of Schreier extensions of monoids with non-abelian kernel
| Autor: | Andrea Montoli, Alex Patchkoria, Nelson Martins-Ferreira, Manuela Sobral |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2020 |
| Předmět: |
Monoid
Pure mathematics Applied Mathematics General Mathematics 010102 general mathematics 01 natural sciences Schreier extension obstruction Eilenberg–Mac Lane cohomology of monoids Kernel (algebra) Obstruction 0103 physical sciences 010307 mathematical physics 0101 mathematics Abelian group Mathematics |
| Zdroj: | Repositório Científico de Acesso Aberto de Portugal Repositório Científico de Acesso Aberto de Portugal (RCAAP) instacron:RCAAP |
| Popis: | 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 ) {\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))} . |
| Databáze: | OpenAIRE |
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