  Let $G$ be a group and let $V$ be an algebraic variety over an algebraically
closed field $K$. Let $A$ denote the set of $K$-points of $V$. We introduce
algebraic sofic subshifts $\Sigma \subset A^G$ and study endomorphisms $\tau
\colon \Sigma \to \Sigma$. We generalize several results for dynamical
invariant sets and nilpotency of $\tau$ that are well known for finite alphabet
cellular automata. Under mild assumptions, we prove that $\tau$ is nilpotent if
and only if its limit set, i.e., the intersection of the images of its
iterates, is a singleton. If moreover $G$ is infinite, finitely generated and
$\Sigma$ is topologically mixing, we show that $\tau$ is nilpotent if and only
if its limit set consists of periodic configurations and has a finite set of
alphabet values.
