For $M\models PA$ let ${\rm{Lt}(M)}=(\{K: K\prec M\},\prec)$ and for $M\prec N$, let ${\rm Lt}(N/M)=(\{K: M\prec K\prec M\},\prec)$.
In general, the lattice problem is: Which lattices can be represented as ${\rm Lt}(N/M)=(\{K: M\prec K\prec M\},\prec)$, for some $M\prec N$?
There is a vast literature on the problem and many special cases remain open. Here are basic references: [1, 2, 3, 4, 5, 6].
Chapter 4 of [7] is devoted to the lattice problem.
Finite lattices
Is every finite lattice lattice a substructure lattice of a model of $PA$?
By a result of Schmerl the answer is positive for all ${\bf M}_n$, where $n=q+1$ or $n=q+2$ and $q$ is a power of a prime. ${\bf M}_n$ is the lattice with a top element, bottom element, and $n$ incomparable elements in between.
The simplest lattice for which the problem is open is ${\bf M}_{16}$.
Every countable $M\models PA$ has an elementary end extension $N$ such that ${\rm Lt}(N/M)$ is isomorphic to the pentagon lattice ${\bf N}_5$ [6], but no $M\models PA$ at all has an elementary end extension such that ${\rm Lt}(N/M) \cong {\bf M}_3$ ([4].
Schmerl has asked: What finite lattices $L$ are such that
every $M\models PA$ has an elementary end extension $N$ such that ${\rm}Lt(N/M)
\cong L$? What finite lattices $L$ are such that every countable $M\models PA$ has an elementary end extension $N$ such that
${\rm Lt}(N/M) \cong L$?
First-order theory of ${\rm Lt}(N/M)$.
Suppose $M_1\prec_{cof} N_1$, $M_2\prec_{cof} N_2$, and $(N_1,M_1)\equiv (N_2,M_2)$. Is ${\rm Lt}(N_1/M_1)$ elementarily equivalent to ${\rm Lt}(N_2/M_2)$? The problem is motivated by a result from [8] showing that the answer is positive for lattices of finitely generated interstructures.
References
- Haim Gaifman. Models and types of Peano's arithmetic. Ann. Math. Logic 9(3):223--306, 1976. MR bibtex
@article {gaifman1976:models,
AUTHOR = {Gaifman, Haim},
TITLE = {Models and types of Peano\'s arithmetic},
JOURNAL = {Ann. Math. Logic},
FJOURNAL = {Annals of Pure and Applied Logic},
VOLUME = {9},
YEAR = {1976},
NUMBER = {3},
PAGES = {223--306},
ISSN = {0168-0072},
MRCLASS = {02H05},
MRNUMBER = {0406791 (53 \#10577)},
MRREVIEWER = {S. R. Kogalovskii},
}
- George Mills. Substructure lattices of models of arithmetic. Ann. Math. Logic 16(2):145--180, 1979. www DOI MR bibtex
@article {mills1979:substructure,
AUTHOR = {Mills, George},
TITLE = {Substructure lattices of models of arithmetic},
JOURNAL = {Ann. Math. Logic},
FJOURNAL = {Annals of Mathematical Logic},
VOLUME = {16},
YEAR = {1979},
NUMBER = {2},
PAGES = {145--180},
ISSN = {0003-4843},
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MRNUMBER = {537207 (81i:03105)},
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- J. B. Paris. On models of arithmetic. Conference in Mathematical Logic---London '70 (Bedford Coll., London, 1970), pp. 251--280. Lecture Notes in Math., Vol. 255, Berlin, 1972. MR bibtex
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AUTHOR = {Paris, J. B.},
TITLE = {On models of arithmetic},
BOOKTITLE = {Conference in Mathematical Logic---London \'70 (Bedford Coll., London, 1970)},
PAGES = {251--280. Lecture Notes in Math., Vol. 255},
PUBLISHER = {Springer},
ADDRESS = {Berlin},
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}
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JOURNAL = {Fund. Math.},
FJOURNAL = {Polska Akademia Nauk. Fundamenta Mathematicae},
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ISSN = {0016-2736},
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- James H. Schmerl. Extending models of arithmetic. Ann. Math. Logic 14:89--109, 1978. www DOI MR bibtex
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JOURNAL = {Ann. Math. Logic},
FJOURNAL = {Annals of Mathematical Logic},
VOLUME = {14},
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PAGES = {89--109},
ISSN = {0003-4843},
CODEN = {AMLOAD},
MRCLASS = {03C62},
MRNUMBER = {506527 (80f:03036)},
MRREVIEWER = {Andreas Blass},
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- Roman Kossak and James H. Schmerl. The structure of models of Peano arithmetic. Vol. 50, The Clarendon Press Oxford University Press, Oxford, 2006. (Oxford Science Publications) www DOI MR bibtex
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TITLE = {The structure of models of Peano arithmetic},
SERIES = {Oxford Logic Guides},
VOLUME = {50},
NOTE = {Oxford Science Publications},
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ADDRESS = {Oxford},
YEAR = {2006},
PAGES = {xiv+311},
ISBN = {978-0-19-856827-8; 0-19-856827-4},
MRCLASS = {03-02 (03C62 03F30 03H15)},
MRNUMBER = {2250469 (2008b:03001)},
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CODEN = {NDJFAM},
MRCLASS = {03C62 (03C62)},
MRNUMBER = {},
MRREVIEWER = {},
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}
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