On constraints and dividing in ternary homogeneous structures

TL;DR

This paper investigates the properties of finitely constrained ternary homogeneous simple structures, establishing conditions for supersimplicity, SU-rank, and dependence, and explores the relationship between constraints, amalgamations, and definable relations, providing new examples.

Contribution

It proves that finitely constrained ternary homogeneous simple structures are supersimple with finite SU-rank and explores their algebraic and definable properties, introducing new examples.

Findings

Finitely constrained structures are supersimple with finite SU-rank.

Algebraic closure is trivial in supersimple structures of SU-rank 1.

New uncountable family of ternary homogeneous supersimple structures of SU-rank 1.

Abstract

Let M be ternary, homogeneous and simple. We prove that if M is finitely constrained, then it is supersimple with finite SU-rank and dependence is $k$-trivial for some $k < \omega$ and for finite sets of real elements. Now suppose that, in addition, M is supersimple with SU-rank 1. If M is finitely constrained then algebraic closure in M is trivial. We also find connections between the nature of the constraints of M, the nature of the amalgamations allowed by the age of M, and the nature of definable equivalence relations. A key method of proof is to "extract" constraints (of 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.