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.