The Pentagon as a Substructure Lattice of Models of Peano Arithmetic

Wilke proved in 1977 that every countable model ${\mathcal M}$ of Peano Arithmetic has an elementary end extension ${\mathcal N}$ such that the interstructure lattice Lt(${\mathcal N} / {\mathcal M}$) is the pentagon lattice ${\mathbf N}_5$. This theorem implies that every countable nonstandard $\mathcal M$ has an elementary cofinal extension such that Lt(${\mathcal N} / {\mathcal M}) \cong {\mathbf N}_5$. It is proved here that if ${\mathcal M} \prec {\mathcal N}$ and Lt(${\mathcal N} / {\mathcal M}) \cong {\mathbf N}_5$, then ${\mathcal N}$ is either an end or a cofinal extension of ${\mathcal M}$. In contrast, there are ${\mathcal M}^* \prec {\mathcal N}^* \models {\mathsf PA}^*$ such that Lt(${\mathcal N} / {\mathcal M}) \cong {\mathbf N}_5$ and ${\mathcal N}^*$ is neither an end nor a cofinal extension of ${\mathcal M}^*$.