Abelian algebras and the hamiltonian property

We show that a finite algebra A is Hamiltonian if the class HS(AA) consists of Abelian algebras. As a consequence, we conclude that a locally finite variety is Abelian if and only if it is Hamiltonian. Furthermore, it is proved that A generates an Abelian variety if and only if AA3 is Hamiltonian. An algebra is Hamiltonian if every nonempty subuniverse is a block of some congruence on the algebra and an algebra is Abelian if for every term t(x,ȳ), the implication t(x, ȳ)=t(x, z̄)→t(w, ȳ)=t(w, z̄) holds. Thus, locally finite Abelian varieties have definable principal congruences, enjoy the congruence extension property, and satisfy the RS-conjecture.