Analogues of Shepherdson's Theorem for the arithmetical language with exponentiation

We investigate the following expansions of $\mathrm{IOpen}$ (Robinson arithmetic Q with quantifier-free induction):

• IOpen(exp): Q with axioms $\mathrm{exp}(0) = 1$ and $\mathrm{exp}(Sx) = exp(x) + exp(x)$ and quantifier-free induction in the expanded language;
• IOpen($x^y$): Q with axioms $x^0 = 1$ and $x^(Sy) = (x^y) \cdot x$ and quantifier-free induction in the expanded language.

In 1964 Shepherdson proved the following theorem characterizing models of IOpen: For every discretely ordered ring $M$, the nonnegative part $M^+$ is a model of IOpen if{f} $M$ is an integer part of the real closure $R(M)$ of the fraction field of $M$ (i.e. for every $r \in R(M)$ exists $m \in M$ such that $m \leqslant r < m + 1$).
Our purpose is to generalize this theorem to the expanded theories IOpen(exp) and IOpen($x^y$). We obtain some partial results for these theories and a full analogue of the Shepherdson's theorem for the theory $\mathrm{IOpen} + T_{x^y}$, where the latter is some finite set of the natural properties of exponentiation.