A dichotomy theorem for regular types

Abstract

Theorem 1. (a) Let T be a superstable theory without the omitting type order property. Then every regular type is either locally modular, or non-orthogonal to a strongly regular type. In the latter case, a realization of the strongly regular type can be found algebraically in any realization of the given one. (b) Let T be a superstable theory with NOTOP and NDOP. Then every regular type is either locally modular or strongly regular. Theorem 2. (a) Let p be a nontrivial regular type. Then p-weight is continuous and definable inside some definable set D of positive p-weight. If p is non-orthogonal to B, then D can be chosen definable over B. (b) Let p be a nontrivial regular type of depth 0. Let stp( a/ B) be p-semi-regular. Then a lies in some acl( B)-definable set D such that p-weight is continuousand definable inside D.