TY - JOUR
T1 - Dichotomy Theorems for Families of Non-Cofinal Essential Complexity
AU - Clemens, John D.
AU - Lecomte, Dominique
AU - Miller, Benjamin D.
N1 - Publisher Copyright:
© 2016 Elsevier Inc.
PY - 2017/1/2
Y1 - 2017/1/2
N2 - We prove that for every Borel equivalence relation E, either E is Borel reducible to E0, or the family of Borel equivalence relations incompatible with E has cofinal essential complexity. It follows that if F is a Borel equivalence relation and F is a family of Borel equivalence relations of non-cofinal essential complexity which together satisfy the dichotomy that for every Borel equivalence relation E, either E∈F or F is Borel reducible to E, then F consists solely of smooth equivalence relations, thus the dichotomy is equivalent to a known theorem.
AB - We prove that for every Borel equivalence relation E, either E is Borel reducible to E0, or the family of Borel equivalence relations incompatible with E has cofinal essential complexity. It follows that if F is a Borel equivalence relation and F is a family of Borel equivalence relations of non-cofinal essential complexity which together satisfy the dichotomy that for every Borel equivalence relation E, either E∈F or F is Borel reducible to E, then F consists solely of smooth equivalence relations, thus the dichotomy is equivalent to a known theorem.
KW - Complexity
KW - Dichotomy
KW - Equivalence relation
KW - Smooth
UR - http://www.scopus.com/inward/record.url?scp=84986880077&partnerID=8YFLogxK
UR - https://scholarworks.boisestate.edu/math_facpubs/189
U2 - 10.1016/j.aim.2016.08.044
DO - 10.1016/j.aim.2016.08.044
M3 - Article
SN - 0001-8708
VL - 304
SP - 285
EP - 299
JO - Advances in Mathematics
JF - Advances in Mathematics
ER -