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