Arithmetical completeness for some extensions of the pure logic of necessitation

We investigate the arithmetical completeness theorems of some extensions of Fitting, Marek, and Truszczy\'{n}ski's pure logic of necessitation $\mathbf{N}$. For $m,n \in \omega$, let $\mathbf{N}^+ \mathbf{A}_{m,n}$, which was introduced by Kurahashi and Sato, be the logic obtained from $\mathbf{N}$ by adding the axiom scheme $\Box^n A \to \Box^m A$ and the rule $\dfrac{\neg \Box A}{\neg \Box \Box A}$. In this paper, among other things, we prove that for each $m,n \geq 1$, the logic $\mathbf{N}^+ \mathbf{A}_{m,n}$ becomes a provability logic.