$2$-affine complete algebras need not be affine complete

Abstract:
Given a finite algebra $\mathbf{A}$, we study the clone $\mathrm{Comp} (\mathbf{A})$ of all congruence preserving functions on $\mathbf{A}$, and the clone $\mathrm{Pol} (\mathbf{A})$ of all polynomial functions on $\mathbf{A}$. We call an algebra $n$-affine complete iff every $n$-ary congruence preserving function is polynomial. For each $k \in \mathbb{N}$, we exhibit an algebra that is $k$-affine complete, but not $(k+1)$-affine complete.