\] We have:  0.  $P_0 \sim_0$ [$S_{0,0}$]{}  0.  $P_0 \sim_0$ [$S_{0,0}$]{}  0.  [$S_{0,0}$]{} $\sim_0$ [$S_{0,0,0}$]{}  0.  [$S_{0,0}$]{} $\sim_0$ [$S_{0,0,0}$]{}.  Therefore, any two elements in ${\mathcal 