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ON CONSTRAINTS AND DIVIDING IN TERNARY HOMOGENEOUS STRUCTURES
Published online by Cambridge University Press: 21 December 2018
Abstract
Let ${\cal M}$ be ternary, homogeneous and simple. We prove that if
${\cal M}$ is finitely constrained, then it is supersimple with finite SU-rank and dependence is k-trivial for some k < ω and for finite sets of real elements. Now suppose that, in addition,
${\cal M}$ is supersimple with SU-rank 1. If
${\cal M}$ is finitely constrained then algebraic closure in
${\cal M}$ is trivial. We also find connections between the nature of the constraints of
${\cal M}$, the nature of the amalgamations allowed by the age of
${\cal M}$, and the nature of definable equivalence relations. A key method of proof is to “extract” constraints (of
${\cal M}$) from instances of dividing and from definable equivalence relations. Finally, we give new examples, including an uncountable family, of ternary homogeneous supersimple structures of SU-rank 1.
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- Copyright © The Association for Symbolic Logic 2018
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