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On sets with rank one in simple homogeneous structures

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Ahlman O., Koponen V.

Fundamenta Mathematicae

2015

tom 228

nr 3

223-250

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 𝓓 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 𝓓 is a reduct of a binary random structure.

Ograniczanie wyników

1 Fundamenta Mathematicae

1 Ahlman O.

1 Koponen V.

1 2015

Wyniki wyszukiwania

On sets with rank one in simple homogeneous structures