Difference between revisions of "Lattices of elementary substructures."
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Most recent: Schmerl, James H. Infinite substructure lattices of models of Peano arithmetic. J. Symbolic Logic 75 (2010), no. 4, 1366–1382. | Most recent: Schmerl, James H. Infinite substructure lattices of models of Peano arithmetic. J. Symbolic Logic 75 (2010), no. 4, 1366–1382. | ||
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| + | == Finite lattices == | ||
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| + | Is every {\cr finite} a lattice substructure lattice? | ||
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| + | 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 ${\bf M}_{16}$. | ||
Revision as of 14:32, 17 January 2013
Let $L$ be a lattice. Is $L$ isomorphic to ${\rm{Lt}(M)}=(\{K: K\prec M\},\prec)$, for some $M\models PA$?
There is a vast literature on the problem and many special cases remain open. Here are basic references:
Paris, J. B. On models of arithmetic. Conference in Mathematical Logic—London '70 (Bedford Coll., London, 1970), pp. 251–280. Lecture Notes in Math., Vol. 255, Springer, Berlin, 1972.
Gaifman, Haim, Models and types of Peano's arithmetic. Ann. Math. Logic 9 (1976), no. 3, 223–306.
Schmerl, James H., Extending models of arithmetic. Ann. Math. Logic 14 (1978), 89–109.
Wilkie, A. J. On models of arithmetic having non-modular substructure lattices. Fund. Math. 95 (1977), no. 3, 223–237.
A chapter of Kossak, Roman; Schmerl, James H. The structure of models of Peano arithmetic. Oxford Logic Guides, 50. Oxford Science Publications. The Clarendon Press, Oxford University Press, Oxford, 2006, is devoted to the lattice problem.
Most recent: Schmerl, James H. Infinite substructure lattices of models of Peano arithmetic. J. Symbolic Logic 75 (2010), no. 4, 1366–1382.
Finite lattices
Is every {\cr finite} a lattice substructure lattice?
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 ${\bf M}_{16}$.
