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 N. For m,n ∈ ω, let N⁺ Am,n, which was introduced by Kurahashi and Sato, be the logic obtained from N by adding the axiom scheme ⁿ A → ᵐ A and the rule A/ A. In this paper, among other things, we prove that for each m,n ≥ 1, the logic N⁺ Am,n becomes a provability logic.