A partial algebra $\A = (A,F)$ is called surjective if each of its elements lies in the range of some of its operations. By a transfinite iteration construction over the class of all ordinals it is proved that in each partial algebra $\A$ there exists the largest surjective subalgebra $\Skr \A$, called the surjective kernel of $\A$. However, what might be found a bit surprising, for each ordinal $\al$ there is an algebra $\A$ with only finitary operations (even with a single unary operation), such that the described construction stops exactly in $\al$ steps. The result is compared with the classical ones on perfect kernels of first countable topological spaces.