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

Abstract

We show that any regular (right) Schreier extension of a monoid M by a monoid A induces an abstract kernel \Phi: M \rightarrow\frac{End(A)}{Inn(A)}. If an abstract kernel factors through \frac{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 coeffcients in the abelian group U(Z(A)) of invertible elements of the center Z(A) of A, on which M acts via \Phi. An abstract kernel \Phi: M \rightarrow\frac{SEnd(A)}{Inn(A)} (resp. \Phi: M \rightarrow\frac{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 \Phi: M \rightarrow\frac{SEnd(A)}{Inn(A)} (resp. \Phi: M \rightarrow\frac{Aut(A)}{Inn(A)}), when it is not empty, is in bijection with the second cohomology group of M with coeffcients in U(Z(A)).