Asymmetric regular types

Abstract

We study asymmetric regular global types p∈S1(C). If p is regular and A-asymmetric then there exists a strict order such that Morley sequences in p over A are strictly increasing (we allow Morley sequences to be indexed by elements of a linear order). We prove that for any small model M ⊇ A maximal Morley sequences in p over A consisting of elements of M have the same (linear) order type, denoted by Invp, A(M). In the countable case we determine all possibilities for Invp, A(M): either it can be any countable linear order, or in any M ⊇ A it is a dense linear order (provided that it has at least two elements). Then we study relationship between Invp, A(M) and Invq, A(M) when p and q are strongly regular, A-asymmetric, and such that p⊇A and q⊇A are not weakly orthogonal. We distinguish two kinds of non-orthogonality: bounded and unbounded. In the bounded case we prove that Invp, A(M) and Invq, A(M) are either isomorphic or anti-isomorphic. In the unbounded case, Invp, A(M) and Invq, A(M) may have distinct cardinalities but we prove that their Dedekind completions are either isomorphic or anti-isomorphic. We provide examples of all four situations.