### $\bold P$-NDOP and $\bold P$-decompositions of $\aleph_\epsilon$-saturated models of superstable theories

by Laskowski and Shelah. [LwSh:933]
Fundamenta Math, 2015
Assume a complete first order theory T is superstable. We generalize revise [Sh:401] in two respects, so do not depend on it. First issue we deal with a more general case. Let bold P be a class of regular types in C, closed under automorphisms and under pm . We generalize [Sh:401] to this context to bold P^pm-saturated M 's, assuming bold P-NDOP which is weaker than NDOP. Second issue, in this content it is more delicate to find sufficient condition on two bold P-decomposition trees to give non-isomorphic models. For this we investigate natural structures on the set of regular types mod pm in M . Actually it suffices to deal with the case M is aleph_epsilon-saturated d_ell = < M^ell_eta,a_eta : eta in I_ell > is a bold P-decomposition of M for ell =1,2 and {p^{d_ell}_eta : eta in I_ell}/ pm = (P cap bold S(M))/ pm and show the two trees are quite similar (or isomorphic).

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