Note on isomorphism theorems of hyperrings.
We analyze semidirect extensions of middle nuclei of commutative automorphic loops. We find a less complicated conditions for the semidirect construction when the middle nucleus is an odd order abelian group. We then use the description to study extensions of orders and .
Let be a commutative groupoid such that ; ; or . Then is determined uniquely up to isomorphism and if it is finite, then for an integer .