An algebra is said to be Abelian if for all terms \(t(x,\bar y)\) and for all a, b, \(\bar c,\) \(\bar d,\) \[ (t(a,\bar c)=t(a,\bar d))\to (t(b,\bar c)=t(b,\bar d)) \] and strongly Abelian if for all terms \(t(x,\bar y)\) and for all a, b, \(\bar c,\) \(\bar d,\) \(\bar e,\) \[ (t(a,\bar c)=t(b,\bar d))\to (t(a,\bar e)=t(b,\bar e)). \] It is Hamiltonian if every nonempty subuniverse is a block of some congruence. A variety \({\mathfrak V}\) is said to have property P if every algebra in \({\mathfrak V}\) has property P. It is not difficult to show that a Hamiltonian variety is Abelian and one would like to know if every locally finite Abelian variety is Hamiltonian. J. Shapiro [Algebra Univers. 25, No.3, 334-364 (1988; Zbl 0654.08001)] showed that a locally finite strongly Abelian variety of algebras with only one fundamental operation is Hamiltonian. The main result of this paper removes the restriction on the number of operators. However, an example is given to show that not all strongly Abelian varieties enjoy this property; thus the local finiteness condition cannot be removed.
In the course of his proof, Shapiro showed that his algebras were quasi- affine, that is, subalgebras of reducts of algebras polynomially equivalent to modules. The final section of this paper gives an example of a five element algebra which is not quasi-affine, but which generates a strongly Abelian variety.