2025/11/08 by VERA KOPONEN, Koponen, Vera
#03C35 #03C45 #03C50 #F.4.1 #FOS: Mathematics #Logic (math.LO)
paper · doi:10.48550/arxiv.2511.06113
For an ω-categorical theory T and model M of T we define a hierarchy of ranks, the n-ranks for n < ω which only care about imaginary elements ``up to level n'', where level n contains every element of M and every imaginary element that is an equivalence class of an ∅-definable equivalence relation on n-tuples of elements from M. Using the n-rank we define the notion of n-independence. For all n < ω, the n-independence relation restricted to Mn has all properties of an independence relation according to Kim and Pillay with the \em possible exception of the symmetry property. We prove that, given any n < ω, if M \models T and the algebraic closure in Meq restricted to imaginary elements ``up to level n'' which have n-rank 1 (over some set of parameters) satisfies the exchange property, then n-independence is symmetric and hence an independence relation when restricted to Mn. Then we show that if n-independence is symmetric for all n < ω, then T is rosy. An application of this is that if T has weak elimination of imaginaries and the algebraic closure in M restricted to elements of M of 0-rank 1 (over some set of parameters from Meq) satisfies the exchange property, then T is superrosy with finite U-thorn-rank.