Let $\mathcal{F}$ be a $P$-filter on $\omega$. Denote by $\Omega=\bigsqcup \omega_i$ where $\omega_i=\omega$. Consider the $P$-filter $\mathcal{S}$ on $\Omega$ whose base is as follows $(\bigsqcup_i F_i, F_i\in \mathcal{F})$.

$\mathcal{F}$ filter is isomorphic to $\mathcal{S}$ filter ?