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Abstract:
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The thesis is a research about nonisolation properties of superstable types over finite domains in general. Two notions of nonisoltions, the notion of eventual-strong (i.e. esn) and the notion of internal are introduced. The thesis consists of three chapters. In Chapter 1 of the thesis the techniques of the stability theory which are used in Chapter 2 and Chapter 3 are overviewed. In Chapter 2 of the thesis NDFC theories are studied and the notions of dimension and U_α-rank through partial orders are developed. It is proved that if the theory T is strictly stable and the the order type of rationals cannot be embedded into the fundamental order of $T$ and there is no strictly stable group interpretable in T^eq, then the theory T has continuum non-isomorphic countable models. It is noted that strongly non-isolated types can be present due to the dimensional discontinuity property. In Chapter 3 of the thesis small superstable theories are studied. In the first part of that chapter the eventual-strong and internally nonisolated types are considered, and some properties were proved. The second part of Chapter 3 contains the proof of the following theorem: if the theory T is a complete, superstable theory, the generic type of every simple group definable in T^eq is orthogonal to all NENI types and sup{U(p)|pϵS(T)}≥ ω^ω holds, then the theory T has continuum non-isomorphic countable models. |