Clones of compatible operations on rings of integers modulo n

A clone on a set $A$ is a set of finitary operations $A^n \to A$ containing all projections and closed under the composition. All clones on the set $A$ form a complete lattice. This lattice is well known in the case $|A| = 2$. If $|A|>2$, then the lattice is uncountable and the reserch is aimed at describing some of its parts.

There are several important clones connected with every algebraic structure $A$. One of them is the clone $P(A)$ of all polynomial operations. Another one is the clone $C(A)$ of all congruence preserving (or compatible) operations. Clearly, $P(A)$ is contained in $C(A)$. If $P(A) = C(A)$, the algebra $A$ is called affine complete. If $A$ is not affine complete, then we are interested in the interval between $P(A)$ and $C(A)$ in the lattice of clones. We consider this problem for rings $\mathbb{Z}_n$ of integers modulo $n$. This ring is affine complete if and only if $n$ is square-free. In general, the problem reduces to the case when $n$ is a prime power. The case $n = p^2$ is also known, the clones $P(\mathbb{Z}_{p^2})$ and $C(\mathbb{Z}_{p^2})$ cover each other. In our talk we solve the problem for $n = p^3$. The interval between $P(\mathbb{Z}_{p^3})$ and $C(\mathbb{Z}_{p^3})$ is (countably) infinite. We describe each clone in this interval both by means of generators and by means of invariant relations. It turns out that the relational description is closely connected with commutator theory.