Asymmetric regular types

We study asymmetric regular global types p is an element of S-1(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 superset of A maximal Morley sequences in p over A consisting of elements of M have the same (linear) order type, denoted by Inv(p,A)(M). In the countable case we determine all possibilities for Inv(p,A)(M): either it can be any countable linear order, or in any M superset of A it is a dense linear order (provided that it has at least two elements). Then we study relationship between Inv(p,A)(M) and Inv(q,A)(M) when p and q are strongly regular, A-asymmetric, and such that p1A and q1A are not weakly orthogonal. We distinguish two kinds of non-orthogonality: bounded and unbounded. In the bounded case we prove that Inv(p,A)(M) and Inv(q,A)(M) are either isomorphic or anti-isomorphic. In the unbounded case, Inv(p,A)(M) and Inv(q,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. (C) 2014 Elsevier B.V. All rights reserved.