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Infinite substructure lattices of models of Peano arithmetic. (English) Zbl 1207.03047

Jeff Paris proved that if \(L\) is a distributive \(\aleph_0\)-algebraic lattice, then every countable model \(M \models \text{PA}\) has an elementary end extension \(N\) such that the lattice of elementary interstructures Lt\((N/M ) = ( \{K : M \preccurlyeq K \preccurlyeq N \}, \preccurlyeq)\) is isomorphic to \(L\) [J. B. Paris, “On models of arithmetic”, Lect. Notes Math. 255, 251–280 (1972; Zbl 0236.02042)]. A modification of Paris’ proof gives a similar result in which \(N\) is a cofinal extension of \(M\).
This is the result that Schmerl improves in a significant way. Day characterized the class of bounded lattices as the smallest class of lattices that contains the one-element lattice and is closed under doubling of intervals [A. Day, “Characterizations of finite lattices that are bounded-homomorphic images or sublattices of free lattices”, Can. J. Math. 31, 69–78 (1979; Zbl 0432.06007)]. Every distributive lattice is bounded, and there are bounded lattices that are not distributive, for example the pentagon lattice N5.
The main theorem of the paper states that the cofinal extension version of Paris’ theorem holds for all bounded \(\aleph_0\)-algebraic lattices. The theorem gives the first example of a model of PA whose lattice of elementary substructures is infinite and finitely generated. The proof rests on a construction of a particular type over a given model of PA, but much preparation is needed that is of purely lattice-theoretic nature. In particular, Schmerl proves a generalization of the celebrated theorem of P. Pudlák and J. Tůma on finite congruence representations of upper bounded lattices, proved in [“Yeast graphs and fermentation of algebraic lattices”, Colloq. Math. Soc. János Bolyai 14, 301–341 (1976; Zbl 0358.06013)].

MSC:

03C62 Models of arithmetic and set theory
03H15 Nonstandard models of arithmetic
06B05 Structure theory of lattices
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