The third cohomology group of a monoid and admissible abstract kernels

We define the product of admissible abstract kernels of the form [Formula: see text], where [Formula: see text] is a monoid, [Formula: see text] is a group and [Formula: see text] is a monoid homomorphism. Identifying [Formula: see text]-equivalent abstract kernels, where [Formula: see text] is the center of [Formula: see text], we obtain that the set [Formula: see text] of [Formula: see text]-equivalence classes of admissible abstract kernels inducing the same action of [Formula: see text] on [Formula: see text] is a commutative monoid. Considering the submonoid [Formula: see text] of abstract kernels that are induced by special Schreier extensions, we prove that the factor monoid [Formula: see text] is an abelian group. Moreover, we show that this abelian group is isomorphic to the third cohomology group [Formula: see text].