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

Abstract

Let $({\omega }^n,⪯)$ be the direct power of $n$ instances of $(\omega ,\le )$, 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 $({\omega }^n,⪯)$ and of $({\omega }^n,\prec )$ have the finite model property and moreover, the modal algebras of the frames $({\omega }^n,⪯)$ and $({\omega }^n,\prec )$ are locally finite.