Lattice rules: projection regularity and unique representations

We introduce a unique characterization for lattice rules which are projection regular. Any such rule, having invariants ${n_1},{n_2}, \ldots ,{n_s}$, may be expressed, uniquely, in the form

$$Qf = \frac{1}{{{n_1}{n_2} \cdots {n_s}}}\sum {\sum \cdots \sum {\... ...}}_2}}}{{{n_2}}} + \cdots + \frac{{{j_s}{{\mathbf{z}}_s}}}{{{n_s}}}} \right)} ,$$

where the matrix $Z = {({{\mathbf{z}}_1},{{\mathbf{z}}_2}, \ldots ,{{\mathbf{z}}_s})^T}$ is upper unit triangular and individual elements satisfy $0 \leq z_r^{(c)} < ({n_r}/{n_c}),r < c$.