2-affine complete algebras need not be affine complete

Abstract.

For each $k \in \mathbb{N}$ we exhibit a finite algebra R _k such that R _k is k-affine complete, but not (k+1)-affine complete; this means that every k-ary congruence preserving function on R _k lies in $\textrm{Pol}_k \textbf{R}_k$, but there is a (k +1)-ary congruence preserving function of R _k that does not lie in $\textrm{Pol}_k \textbf{R}_k$.