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 \[ Qf = \frac {1}{{{n_1}{n_2} \cdots {n_s}}}\sum {\sum \cdots \sum {\bar f} \left ( {\frac {{{j_1}{{\mathbf {z}}_1}}}{{{n_1}}} + \frac {{{j_2}{{\mathbf {z}}_2}}}{{{n_2}}} + \cdots + \frac {{{j_s}{{\mathbf {z}}_s}}}{{{n_s}}}} \right )} ,\] where the matrix $Z = {({{\mathbf {z}}_1},{{\mathbf {z}}_2}, \ldots ,{{\mathbf {z}}_s})^T}$ is upper unit triangular and individual elements satisfy $0 \leq z_r^{(c)} < ({n_r}/{n_c}),r < c$.