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Publication
On the classification of Schreier extensions of monoids with non-abelian kernel
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Advisor(s)
Abstract(s)
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) . If an abstract kernel factors through 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 coefficients in the abelian group U(Z(A)) of invertible elements of the center Z(A)
of A, on which M acts via Φ. An abstract kernel Φ: M → SEnd(A)/Inn(A) (resp. Φ: M → 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 Φ: M → SEnd(A)/Inn(A) (resp. Φ: M → Aut(A)/Inn(A) ), when it is not empty, is in bijection with the second cohomology group of M with coefficients in U(Z(A)).
Description
This work was partially supported by the Centre for Mathematics of the University of Coimbra –
UID/MAT/00324/2019, by ESTG and CDRSP from the Polytechnical Institute of Leiria – UID/Multi/04044/
2019, funded by the Portuguese Government through FCT/MCTES and co-funded by the European Regional
Development Fund through the Partnership Agreement PT2020. The second author was partially supported
by the Programma per Giovani Ricercatori “Rita Levi-Montalcini”, funded by the Italian government through
MIUR. The third author was supported by the Shota Rustaveli National Science Foundation of Georgia
(SRNSFG), grant FR-18-10849, “Stable Structures in Homological Algebra”.
Keywords
Monoid Schreier extension Obstruction Eilenberg–Mac Lane cohomology of monoids
Citation
Martins-Ferreira, N., Montoli, A., Patchkoria, A. & Sobral, M. (2020). On the classification of Schreier extensions of monoids with non-abelian kernel. Forum Mathematicum, 32(3), 607-623. https://doi.org/10.1515/forum-2019-0164
Publisher
De Gruyter