Operators are defined which generate a sublattice of a given Penrose lattice. The different Penrose lattices as well as further non-periodic lattices are sublattices of other Penrose lattices. The procedure of inflation is equivalent to a special sublattice operator. Combinations of operators produce new operators; the according rules are described as operator multiplication. Novel non-periodic lattices deduced by the action of an operator on a Penrose lattice are demonstrated.