Data di Pubblicazione:
2020
Citazione:
On the classification of Schreier extensions of monoids with non-abelian kernel / N. Martins-Ferreira, A. Montoli, A. Patchkoria, M. Sobral. - In: FORUM MATHEMATICUM. - ISSN 0933-7741. - 32:3(2020), pp. 607-623. [10.1515/forum-2019-0164]
Abstract:
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) Phicolon Mtofrac 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) Phicolon Mtofrac operatorname SEnd (A) operatorname Inn (A) (resp. φ: M → Aut (A) Inn (A) Phicolon Mtofrac 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) Phicolon Mtofrac operatorname SEnd (A) operatorname Inn (A) (resp. φ: M → Aut (A) Inn (A) Phicolon Mtofrac 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)) .
Tipologia IRIS:
01 - Articolo su periodico
Keywords:
Eilenberg-Mac Lane cohomology of monoids; Monoid; obstruction; Schreier extension
Elenco autori:
N. Martins-Ferreira, A. Montoli, A. Patchkoria, M. Sobral
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