Lattice substitution systems and model sets

This paper studies ways in which the sets of a partition of a lattice in ℝn become regular model sets. The main theorem gives equivalent conditions which assure that a matrix substitution system on a lattice in ℝn gives rise to regular model sets (based on p-adic-like internal spaces), and hence to pure point diffractive sets. The methods developed here are used to show that the n-dimensional chair tiling and the sphinx tiling are pure point diffractive.