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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 $(\omega^n,\prec)$ havethe finite model property and moreover, the modal algebras of the frames$(\omega^n,\preceq)$ and $(\omega^n,\prec)$ are locally finite.

Modal logics of finite direct powers of $\omega$ have the finite model property