On the classification of Schreier extensions of monoids with non-abelian kernel

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)) .