A dichotomy for $D$-rank 1 types in simple theories

Ziv Shami

Published 2013-11-16, updated 2019-09-17Version 2

We prove a dichotomy for $D$-rank 1 types in simple theories that generalizes Buechler's dichotomy for $D$-rank 1 minimal types: every such type is either 1-based or its algebraic closure, by a single formula, almost contains a non-algebraic formula that belongs to a non-forking extension of the type. In addition we prove that a densely-1 based type of $D$-rank 1 is 1-based. We also observe that for a hypersimple unidimensional theory the existence of a non-algebraic stable type implies stability (and thus superstability).