  Solution sets of systems of homogeneous linear equations over fields are
characterized as being subspaces, i.e., sets that are closed under linear
combinations. Our goal is to characterize solution sets of systems of equations
over arbitrary finite algebras by a similar closure condition. We show that
solution sets are always closed under the centralizer of the clone of term
operations of the given algebra; moreover, the centralizer is the only clone
that could characterize solution sets. If every centralizer-closed set is the
set of all solutions of a system of equations over a finite algebra, then we
say that the algebra has Property (SDC). Our main result is the description of
finite lattices and semilattices with Property (SDC): we prove that a finite
lattice has Property (SDC) if and only if it is a Boolean lattice, and a finite
semilattice has Property (SDC) if and only if it is distributive.
