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

## TL;DR

This paper proves that the modal logics of finite direct powers of natural numbers with standard and strict orderings have the finite model property, and their modal algebras are locally finite, advancing understanding of their logical structure.

## Contribution

It establishes the finite model property and local finiteness of modal algebras for frames based on finite powers of 0 with standard and strict orderings, which was previously unknown.

## Key findings

- Modal logics of (0^n,\u2261) have the finite model property.
- Modal logics of (0^n,<) have the finite model property.
- Modal algebras of these frames are locally finite.

## Abstract

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.