  We study the finite frame property of some extensions of Fitting, Marek, and
Truszczy\'nski's pure logic of necessitation $\mathbf{N}$. For any natural
numbers $m, n$, we introduce the logic $\mathbf{N}^+\mathbf{A}_{m, n}$ by
adding the single axiom scheme $\Box^n \varphi \to \Box^m \varphi$ and the rule
$\dfrac{\neg \Box \varphi}{\neg \Box \Box \varphi}$ (Ros$^\Box$) into
$\mathbf{N}$. We prove the finite frame property of $\mathbf{N}^+\mathbf{A}_{m,
n}$ with respect to Fitting, Marek, and Truszczy\'nski's relational semantics.
We also prove that for $n \ge 2$, the logic obtained by removing the rule
Ros$^\Box$ from $\mathbf{N}^+\mathbf{A}_{0, n}$ is incomplete with respect to
that semantics.
