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Abstract:
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In this thesis we study asymmetric regular types. If p is regular and
asymmetric over A, then there exists an order such that Morley sequences in p over
A are strictly increasing. It turns out that for every small model M A, the order
type of a maximal Morley sequence in p over A whose elements are from M does not
depend on the choice of the sequence, i.e. it is an invariant of the model M denoted
by Invp;A(M). In the countable case we can determine all possibilities for Invp;A(M):
either Invp;A(M) is an arbitrary countable linear order or, provided that it contains
at least two elements, it is a countable dense linear order (possibly with one or
both endpoints). Also, we study the connection between Invp;A(M) and Invq;A(M),
where p and q are two regular and asymmetric over A types such that p A 6?w q A. We
distinguish two kinds of non-orthogonality: bounded and unbounded. Under the
assumption that p and q are convex, in the bounded case we get that Invp;A(M) and
Invq;A(M) are either isomorphic or anti-isomorphic, while under the assumption
of strong regularity, in the unbounded case we get that Dedekind completions of
Invp;A(M) and Invq;A(M) are either isomorphic or anti-isomorphic.
In particular we study the following class of structures: expansions of linear
orderings with countably many unary predicates and countably many equivalence
relations with convex classes. We provide new examples of regular types. Namely,
it turns out that every global invariant type in this context is regular, and every
non-algebraic type over A has precisely two global extensions which are invariant
over A.
We also study the connection between the question of existence of a quasi-
minimal model of a complete rst-order theory and the question of existence of
a global strongly regular type. We also deal with the problem whether every quasi-
minimal group must be abelian. It turns out that this question has the positive
answer provided that the global extension of the generic type of a quasi-minimal
group is asymmetric over ;. |