Models of quantifier-free induction for the language of arithmetic with exponentiation

In 1964 Shepherdson proved the following fact: a discretely ordered semiring $M^+$ satisfies $\mathrm{IOpen}$ (the induction schema for quantifier-free formulas) iff the corresponding ring $M$ is the integer part of the real closure of the quotient field of $M$. ($M$ is called an integer part of $R$ if $M$ is discretely ordered and for all $r$ in $R$ there exists an $m$ in $M$ such that $m \leq r < m+1$.) We consider the expansions of $\mathrm{IOpen}$ with the exponentiation and the power functions and try to find similar criteria to build models of these theories.
The talk will be in Russian.