Fix a countable nonstandard model $mathcal M$ of Peano Arithmetic. Even with some rather severe restrictions placed on the types of minimal cofinal extensions $mathcal N succ mathcal M$ that are allowed, we still find that there are $2^{aleph_0}$ pos
sible theories of $(mathcal N,M)$ for such $mathcal N$s.
If $M prec N$ are models of Peano Arithmetic and Lt$(N/M)$ is the pentagon lattice $N_5$, then $N$ is either a cofinal or an end extension of $M$. In contrast, there are $M prec N$ that are models of PA* (PA in a language with countably many new pred
icate symbols) such that Lt$(N/M) cong N_5$ and $N$ is neither a cofinal nor an end extension of $M$.
