On the existence of regular types

The main results in the paper are the following. Theorem A. Suppose that T is superstable and M ⊂ N are distinct models of T^eq. Then there is a c ε{lunate} N{minus 45 degree rule}M such that t( c M) is regular. For M ⊂ N two models we say that M ⊂_naN if for all a ε{lunate} M and θ(x, a) such that θ(M, a) ≠ θ(N, a), there is a b ∈ θ(M, a) {minus 45 degree rule}acl(a). Theorem BSuppose that T is superstable, M ⊂_naN are models of T^eq, and p is a regular type non-orthogonal to t( N M). Then there is a c ε{lunate} N such that t( c M) is regular and non-orthogonal to p. Furthermore, there is a formula θ ∈ t( c M) such that a ∈ θ(N) and t( a M)⊥ {combining short solidus overlay} p ⇒ t( a M) is regular. We used these results to obtain 'good' tree decompositions of models in (possibly uncountable) superstable theories with NDOP. See Definition 5.1 for the undefined terms. Theorem C Suppose that T is superstable with NDOP and M {true} T^eq. Then every ⊂_na- decomposition inside M extends to a ⊂_na-decomposition of M. Furthermore, if 〈N_η, a_η: η ∈ I〉 is any ⊂_na-decomposition of M, then M is minimal over ∪N_η and for all η ∈ I. M is dominated by ∪N_η over N_η. Using some stable group theory we show that when Th(M) is superstable with NDOP and 〈N_η: η ∈ I〉 is a tree decomposition of M, then M is constructible over ∪ n_η with respect to a very strong isolation relation (Section 6).