On sets with rank one in simple homogeneous structures

Ove Ahlman ¹   ; Vera Koponen ¹

¹ Department of Mathematics Uppsala University Box 480 75106 Uppsala, Sweden

Fundamenta Mathematicae, Tome 228 (2015) no. 3, pp. 223-250

Résumé

We study definable sets $D$ of SU-rank 1 in $\mathcal M^{\rm eq}$, where $\mathcal M$ is a countable homogeneous and simple structure in a language with finite relational vocabulary. Each such $D$ can be seen as a `canonically embedded structure', which inherits all relations on $D$ which are definable in $\mathcal M^{\rm eq}$, and has no other definable relations. Our results imply that if no relation symbol of the language of $\mathcal M$ has arity higher than 2, then there is a close relationship between triviality of dependence and $\mathcal D$ being a reduct of a binary random structure. Somewhat more precisely: (a) if for every $n \geq 2$, every $n$-type $p(x_1, \ldots , x_n)$ which is realized in $D$ is determined by its sub-2-types $q(x_i, x_j) \subseteq p$, then the algebraic closure restricted to $D$ is trivial; (b) if $\mathcal M$ has trivial dependence, then $\mathcal D$ is a reduct of a binary random structure.

Détail

DOI : 10.4064/fm228-3-2

Keywords: study definable sets su rank mathcal where mathcal countable homogeneous simple structure language finite relational vocabulary each seen canonically embedded structure which inherits relations nbsp which definable mathcal has other definable relations results imply relation symbol language mathcal has arity higher nbsp there close relationship between triviality dependence mathcal being reduct binary random structure somewhat precisely every geq every n type ldots which realized determined its sub types subseteq algebraic closure restricted trivial mathcal has trivial dependence mathcal reduct binary random structure