We study definable sets D of SU-rank 1 in , where 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 , and has no other definable relations. Our results imply that if no relation symbol of the language of ℳ has arity higher than 2, then there is a close relationship between triviality of dependence and being a reduct of a binary random structure. Somewhat more precisely: (a) if for every n ≥ 2, every n-type p(x₁, ..., xₙ) which is realized in D is determined by its sub-2-types , then the algebraic closure restricted to D is trivial; (b) if ℳ has trivial dependence, then is a reduct of a binary random structure.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm228-3-2,
author = {Ove Ahlman and Vera Koponen},
title = {On sets with rank one in simple homogeneous structures},
journal = {Fundamenta Mathematicae},
volume = {228},
year = {2015},
pages = {223-250},
zbl = {06390220},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm228-3-2}
}
Ove Ahlman; Vera Koponen. On sets with rank one in simple homogeneous structures. Fundamenta Mathematicae, Tome 228 (2015) pp. 223-250. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm228-3-2/