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Bulletin of Irkutsk State University. Series Mathematics, 2014, Volume 9, Pages 118–133 (Mi iigum204)  

On Existence of Limit Models over Sequences of Types

S. V. Sudoplatovabc

a Sobolev Institute of Mathematics SB RAS, 4, Academician Koptyug Avenue, Novosibirsk, 630090
b Novosibirsk State Technical University, 20, K. Marx Avenue, Novosibirsk, 630073
c Novosibirsk State University, 2, Pirogova st., Novosibirsk, 630090
References:
Abstract: 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 $\omega$.
Keywords: limit model, sequence of types, Morley rank.
Document Type: Article
UDC: 510.67
Language: Russian
Citation: S. V. Sudoplatov, “On Existence of Limit Models over Sequences of Types”, Bulletin of Irkutsk State University. Series Mathematics, 9 (2014), 118–133
Citation in format AMSBIB
\Bibitem{Sud14}
\by S.~V.~Sudoplatov
\paper On Existence of Limit Models over Sequences of Types
\jour Bulletin of Irkutsk State University. Series Mathematics
\yr 2014
\vol 9
\pages 118--133
\mathnet{http://mi.mathnet.ru/iigum204}
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