Geodesic
Logical theories of one-place functions on the set of natural numbers
Izvestiya. Mathematics , Tome 22 (1984) no. 3, pp. 587-618

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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 $\langle\mathbf N;+,f\rangle$. Bibliography: 20 titles. 
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     author = {A. L. Semenov},
     title = {Logical theories of one-place functions on the set of natural numbers},
     journal = {Izvestiya. Mathematics },
     pages = {587--618},
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     number = {3},
     year = {1984},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1984_22_3_a5/}
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A. L. Semenov. Logical theories of one-place functions on the set of natural numbers. Izvestiya. Mathematics , Tome 22 (1984) no. 3, pp. 587-618. http://geodesic.mathdoc.fr/item/IM2_1984_22_3_a5/