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Every countable $M\models PA$ has an elementary end extension $N$ such that ${\rm Lt}(N/M)$ is isomorphic to the pentagon lattice ${\bf N}_5$ <cite> wilkie1977:onmodels </cite>, but no $M\models PA$ at all has an elementary end extension such that ${\rm Lt}(N/M) \cong {\bf M}_3$ (<cite> paris1977:modelsof </cite>.
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Every countable $M\models PA$ has an elementary end extension $N$ such that ${\rm Lt}(N/M)$ is isomorphic to the pentagon lattice ${\bf N}_5$ <cite> wilkie1977:onmodels </cite>, but no $M\models PA$ at all has an elementary end extension such that ${\rm Lt}(N/M) \cong {\bf M}_3$ <cite> paris1977:modelsof </cite>.
    
Schmerl has asked:  What finite lattices $L$  are such that
 
Schmerl has asked:  What finite lattices $L$  are such that