Аннотация:In this paper the author studies the decision problem for logical languages intended to describe the properties of one-place functions f on the set N of natural numbers. For functions f taking a finite number of values a criterion for decidability of the monadic theory of the structure ⟨N;<=,f⟩ is obtained. For a large class of monotone functions f, conditions are found under which the elementary theory of the structure ⟨N;<=,f⟩ is decidable; corresponding conditions are also found for structures of the form ⟨N;+,f⟩.