Solution sets of systems of equations over finite lattices and semilattices

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 $${{\,\mathrm{(SDC)}\,}}$$. Our main result is the description of finite lattices and semilattices with Property $${{\,\mathrm{(SDC)}\,}}$$: we prove that a finite lattice has Property $${{\,\mathrm{(SDC)}\,}}$$ if and only if it is a Boolean lattice, and a finite semilattice has Property $${{\,\mathrm{(SDC)}\,}}$$ if and only if it is distributive.