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

Abstract

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 ) {\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))} .