  Let $(\omega^n,\preceq)$ be the direct power of $n$ instances of
$(\omega,\leq)$, natural numbers with the standard ordering, $(\omega^n,\prec)$
the direct power of $n$ instances of $(\omega,<)$. We show that for all finite
$n$, the modal logics of $(\omega^n,\preceq)$ and of $(\omega^n,\prec)$ have
the finite model property and moreover, the modal algebras of the frames
$(\omega^n,\preceq)$ and $(\omega^n,\prec)$ are locally finite.
