On Existence of Limit Models over Sequences of Types

S. V. Sudoplatov

The Bulletin of Irkutsk State University. Series Mathematics, Tome 9 (2014), pp. 118-133

Résumé

We consider limit models, i.e., countable models representable as unions of elementary chains of prime models over finite sets, but not isomorphic to any prime model over a finite set. Any countable model of small theory (i.e., of theory with countably many types) is either prime over a tuple or limit. Moreover, any limit model is either limit over a type, i.e., can be represented as a union of elementary chain of pairwise isomorphic prime models over realizations of some fixed type, or limit over a sequence of pairwise distinct types, over which prime models are not isomorphic. In the paper, we characterize the property of existence of limit model over a sequence of types in terms of relations of isolation and semi-isolation: it is shown that there is a limit model over a sequence of types if and only if there are infinitely many non-symmetric transitions between types with respect to relation of isolation, or, that is equivalent, with respect to relation of semi-isolation. These criteria generalize the related criteria for limit models over a type. We characterize, in terms of relations of isolation and semi-isolation, the condition of existence of a limit model over a subsequence of a given sequence of types. We prove that if a theory has a limit model over a type then the Morley rank of this theory is infinite. Moreover, some restriction of the theory to some finite language has infinite Morley rank. That estimation is precise: there is an $\omega$-stable theory with a limit model over a type and having Morley rank $\omega$.

Détail

Keywords: limit model, sequence of types, Morley rank.

Bibliographie

[1] Sudoplatov S. V., The Lachlan problem, NSTU, Novosibirsk, 2009, 336 pp. (in Russian)

[2] Sudoplatov S. V., “Hypergraphs of prime models and distributions of countable models of small theories”, J. Math. Sciences, 169:5 (2010), 680–695 | DOI | MR | Zbl

[3] B. S. Baizhanov, S. V. Sudoplatov, V. V. Verbovskiy, “Conditions for non-symmetric relations of semi-isolation”, Siberian Electronic Mathematical Reports, 9 (2012), 161–184 | MR

[4] Sudoplatov S. V., “Complete theories with finitely many countable models, I”, Algebra and Logic, 43:1 (2004), 62–69 | DOI | MR | Zbl

[5] J. T. Baldwin, A. H. Lachlan, “On strongly minimal sets”, J. Symbolic Logic, 36:1 (1971), 79–96 | DOI | MR | Zbl

[6] A. Pillay, A note on one-based theories, Preprint, University of Notre Dame, Notre Dame, 1989, 5 pp.

[7] Sudoplatov S. V., “On powerful types in small Theories”, Siberian Math. J., 31:4 (1990), 629–638 | DOI | MR | Zbl

[8] Sudoplatov S. V., “On limit models over types in the class of $\omega$-stable theories”, Reports of Irkutsk State University. Series: Mathematics, 3:4 (2010), 114–120 (in Russian) | Zbl

[9] A. Pillay, “Countable models of stable theories”, Proc. Amer. Math. Soc., 89:4 (1983), 666–672 | DOI | MR | Zbl

[10] E. Casanovas, The number of countable models, Preprint, University of Barcelona, Barcelona, 2012, 19 pp.

[11] S. V. Sudoplatov, “On Rudin–Keisler preorders in small theories”, Algebra and Model Theory 8, Collection of paperseds. . A. G. Pinus, K. N. Ponomaryov, S. V. Sudoplatov, E. I. Timoshenko, NSTU, Novosibirsk, 2011, 94–102

[12] B. Kim, “On the number of countable models of a countable supersimple theory”, J. London Math. Soc., 60:2 (1999), 641–645 | DOI | MR | Zbl

[13] P. Tanović, “Theories with constants and three countable models”, Archive for Math. Logic, 46:5–6 (2007), 517–527 | DOI | MR | Zbl

[14] P. Tanović, “Asymmetric $\mathrm{RK}$-minimal types”, Archive for Math. Logic, 49:3 (2010), 367–377 | DOI | MR | Zbl