Arithmetic Interpretability Types of Varieties and Some Additive Problems with Primes

We deal with varieties with one basic operation $f(x_1,\dots,x_n)$ and one defining identity $f(x_1,\dots,x_n)=f(x_\pi(1),\dots, x_\pi(n))$, where $\pi$ is a permutation whose cyclic set consists of distinct primes $p_1,\dots, p_r$, with the sum $p_1+\dots+p_r=n$. Their interpretability types, together with the greatest element $\mathbf1$ in a lattice $\mathbb L^\mathrm{int}$, are said to be arithmetic. It is proved that the arithmetic types constitute a distributive lattice $\mathbb L_\mathrm{ar}$, which is dual to a lattice $\mathrm{Sub}_f\Pi$ of finite subsets of the set $\Pi$ of all primes. It is shown that for $n\geqslant2$, the poset $\mathbb L_\mathrm{ar}(\mathbb S_n)$ of arithmetic types defined by permutations in $\mathbb S_n$, for $n$ fixed, is a lattice iff $n=2,3,4,6,8,9,11$.