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.