Finitely generated variety

Remarks

If $\mathcal{K} = \set{A_1, \dots, A_k}$, then $\mathbf{HSP}(\mathcal{K}) = \mathbf{HSP}(A_1 \times \cdots \times A_k)$, so one finite algebra suffices. Example: the variety of abelian groups that satisfy the identity $6x = 0$ is generated by the finite group $\Z/6\Z$. Each such group is a direct sum of copies of $\Z/2\Z$ and $\Z/3\Z$, and both of these are subgroups of $\Z/6\Z$. A direct sum is a subgroup of the direct product, so each such group lies in $\mathbf{SP}(\Z/6\Z)$. Non-example: the variety of all abelian groups is not finitely generated. If $A$ is a finite abelian group with $m$ elements, then every group in $\mathbf{HSP}(A)$ satisfies the identity $mx = 0$, while $\Z$ does not. Every finitely generated variety is a locally finite variety.