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