Logical theories of one-place functions on the set of natural numbers

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 $\mathbf N$ of natural numbers. For functions $f$ taking a finite number of values a criterion for decidability of the monadic theory of the structure $\langle\mathbf N;\leqslant,f\rangle$ is obtained. For a large class of monotone functions $f$, conditions are found under which the elementary theory of the structure $\langle\mathbf N;\leqslant,f\rangle$ is decidable; corresponding conditions are also found for structures of the form $\langle\mathbf N;+,f\rangle$.
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