The aim of this paper is twofold. First some machinery is established to reveal the structure of abelian congruences. Then we describe all minimal, locally finite, locally solvable varieties. For locally solvable varieties, this solves problems 9 and 10 of Hobby and McKenzie. We generalize part of this result by proving that all locally finite varieties generated by nilpotent algebras that have...

Using an old example of Nagata, we construct a Noetherian ring prime characteristic p, whose Frobenius morphism is locally finite, but not finite.