Arithmetical completeness for some extensions of the pure logic of necessitation

We investigate the arithmetical completeness theorems of some extensions of Fitting, Marek, and Truszczyński’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 ${\mathrm{\square }}^{n}A\to {\mathrm{\square }}^{m}A$. In this paper, among other things, we prove that for each $m,n\ge 1$, the logic ${\mathbf{N}\mathbf{A}}_{m,n}$ becomes a provability logic, that is, there exists a provability predicate ${\mathrm{P}\mathrm{r}}_T(x)$ of $T$ whose $T$-verifiable modal principles are exactly the logic ${\mathbf{N}\mathbf{A}}_{m,n}$.