Geodesic
On certain extensions of the arithmetic of addition of natural numbers
Izvestiya. Mathematics , Tome 15 (1980) no. 2, pp. 401-418

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In this paper the problems of expressibility and decidability are studied for elementary theories obtained by extending the arithmetic of order and the arithmetic of addition of natural numbers. Results are obtained on the decidability and undecidability of elementary theories of concrete structures of the form $\langle\mathbf N;+,P\rangle$, where $P$ is a fixed monadic predicate, as well as results on the class of sets definable in the theory $\mathbf T\langle\mathbf N;+,\lambda x,\exists\,y\,(x=d^y)\rangle$. Bibliography: 6 titles. 
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     author = {A. L. Semenov},
     title = {On certain extensions of the arithmetic of addition of natural numbers},
     journal = {Izvestiya. Mathematics },
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A. L. Semenov. On certain extensions of the arithmetic of addition of natural numbers. Izvestiya. Mathematics , Tome 15 (1980) no. 2, pp. 401-418. http://geodesic.mathdoc.fr/item/IM2_1980_15_2_a8/