On Existence of Limit Models over Sequences of Types
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 ω-stable theory with a limit model over a type and having Morley rank ω.
1. Baizhanov B.S., Sudoplatov S.V., Verbovskiy V.V. Conditions for non-symmetric relations of semi-isolation. Siberian Electronic Mathematical Reports, 2012, vol. 9, pp. 161-184.
2. Baldwin J.T., Lachlan A.H. On strongly minimal Sets. J. Symbolic Logic, 1971, vol. 36, no 1, pp. 79-96.
3. Casanovas E. The number of countable models. Barcelona, University of Barcelona, 2012, 19 p. (Preprint).
4. Kim B. On the number of countable models of a countable supersimple theory. J. London Math. Soc., 1999, vol. 60, no 2, pp. 641-645.
5. Pillay A. Countable models of stable theories. Proc. Amer. Math. Soc., 1983, vol. 89, no 4, pp. 666-672.
6. Pillay A. A note on one-based theories. Notre Dame, University of Notre Dame, 1989, 5 p. (Preprint).
7. Sudoplatov S.V. On powerful types in small Theories. Siberian Math. J., 1990, vol. 31, no 4, pp. 629-638.
8. Sudoplatov S.V. Complete theories with finitely many countable models. I. Algebra and Logic, 2004, vol. 43, no 1, pp. 62-69.
9. Sudoplatov S.V. The Lachlan problem. Novosibirsk, NSTU, 2009, 336 p. [in Russian]
10. Sudoplatov S.V. Hypergraphs of prime models and distributions of countable models of small theories. J. Math. Sciences, 2010, vol. 169, no 5, pp. 680-695.
11. Sudoplatov S.V. On limit models over types in the class of ω-stable theories. Reports of Irkutsk State University. Series: Mathematics, 2010, vol. 3, no 4, pp. 114-120. [in Russian]
12. Sudoplatov S. V. On Rudin–Keisler preorders in small theories. Algebra and Model Theory 8. Collection of papers, eds.: A.G. Pinus, K.N. Ponomaryov, S.V. Sudoplatov and E.I. Timoshenko, Novosibirsk : NSTU, 2011, pp. 94-102.
13. Tanovic P. Theories with constants and three countable models. Archive for Math. Logic, 2007, vol. 46, no 5-6, p. 517-527.
14. Tanovic P. Asymmetric RK-minimal types. Archive for Math. Logic., 2010, vol. 49, no 3, pp. 367-377.