TY - JOUR
T1 - A dichotomy for D-rank 1 types in simple theories
AU - Shami, Ziv
N1 - Publisher Copyright:
© 2015, Hebrew University of Jerusalem.
PY - 2015/9/1
Y1 - 2015/9/1
N2 - We prove a dichotomy for D-rank 1 types in simple theories that generalizes Buechler’s dichotomy for D-rank 1 minimal types in stable theories: every D-rank 1 type is either 1-based or part of its algebraic closure, defined 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).
AB - We prove a dichotomy for D-rank 1 types in simple theories that generalizes Buechler’s dichotomy for D-rank 1 minimal types in stable theories: every D-rank 1 type is either 1-based or part of its algebraic closure, defined 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).
UR - http://www.scopus.com/inward/record.url?scp=84945900050&partnerID=8YFLogxK
U2 - 10.1007/s11856-015-1243-z
DO - 10.1007/s11856-015-1243-z
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AN - SCOPUS:84945900050
SN - 0021-2172
VL - 209
SP - 993
EP - 1012
JO - Israel Journal of Mathematics
JF - Israel Journal of Mathematics
IS - 2
ER -