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\'nski's pure logic of necessitation N. For m, n, let N^+ A₌, ₍, 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^+ A₌, ₍ becomes a provability logic.