We show that any regular (right) Schreier extension of a monoid M by a monoid A induces an abstract kernel .
If an abstract kernel factors through , where 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 of invertible elements of the center of A, on which M acts via Φ.
An abstract kernel (resp. ) 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 (resp. ), when it is not empty, is in bijection with the second cohomology group of M with coefficients in .
N. Martins-Ferreira, A. Montoli and M. Sobral,
The Nine Lemma and the push forward construction for special Schreier extensions of monoids with operations,
Semigroup Forum 97 (2018), 325–352.
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”.
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