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ON CONSTRAINTS AND DIVIDING IN TERNARY HOMOGENEOUS STRUCTURES

Published online by Cambridge University Press:  21 December 2018

VERA KOPONEN
VERA KOPONEN*
Affiliation:
DEPARTMENT OF MATHEMATICS UPPSALA UNIVERSITY, BOX 480, 75106UPPSALA, SWEDENE-mail: [email protected]
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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.

Keywords

Primary 03C10Secondary 03C4503C1303C1503C30model theoryhomogeneous structuresimple theoryconstraintdividingamalgamation
Type
Articles
Information
The Journal of Symbolic Logic , Volume 83 , Issue 4 , December 2018 , pp. 1691 - 1721
Copyright
Copyright © The Association for Symbolic Logic 2018 

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