On sets with rank one in simple homogeneous structures

Ove Ahlman, and Vera Koponen. "On sets with rank one in simple homogeneous structures." Fundamenta Mathematicae 228.3 (2015): 223-250. <http://eudml.org/doc/286641>.

@article{OveAhlman2015,
abstract = {We study definable sets D of SU-rank 1 in $ℳ^\{eq\}$, 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 $ℳ^\{eq\}$, 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 $q(x_\{i\},x_\{j\}) ⊆ p$, then the algebraic closure restricted to D is trivial; (b) if ℳ has trivial dependence, then is a reduct of a binary random structure.},
author = {Ove Ahlman, Vera Koponen},
journal = {Fundamenta Mathematicae},
keywords = {model theory; homogeneous structure; simple theory; pregeometry; rank; reduct; random structure},
language = {eng},
number = {3},
pages = {223-250},
title = {On sets with rank one in simple homogeneous structures},
url = {http://eudml.org/doc/286641},
volume = {228},
year = {2015},
}

TY - JOUR
AU - Ove Ahlman
AU - Vera Koponen
TI - On sets with rank one in simple homogeneous structures
JO - Fundamenta Mathematicae
PY - 2015
VL - 228
IS - 3
SP - 223
EP - 250
AB - We study definable sets D of SU-rank 1 in $ℳ^{eq}$, 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 $ℳ^{eq}$, 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 $q(x_{i},x_{j}) ⊆ p$, then the algebraic closure restricted to D is trivial; (b) if ℳ has trivial dependence, then is a reduct of a binary random structure.
LA - eng
KW - model theory; homogeneous structure; simple theory; pregeometry; rank; reduct; random structure
UR - http://eudml.org/doc/286641
ER -