Varieties Defined by Permutations

We continue to study interrelations between permutative varieties and the cyclic varieties defined by cycles of the form $(1\,2\ldots k)$. A criterion is given determining whether a cyclic variety $G_k$ is interpretable in ${}_nG_\pi$. For a permutation $\pi$ without fixed elements, it is stated that a set of primes $p$ for which ${}_nG_\pi$ is interpretable in $G_p$ in the lattice $\mathbb L^{\rm int}$ is finite. It is also proved that for distinct primes $p_1,\ldots,p_r$, the Helly number of a type $[G_{p_1}]\wedge\ldots\wedge[G_{p_r}]$ in $\mathbb L^{\rm int}$ coincides with dimension of the dual type $[G_{p_1}]\vee\ldots\vee[G_{p_r}]$ and equals $r$.