Notions of rank and independence in countably categorical theories

TL;DR

This paper introduces a hierarchy of ranks and independence notions in countably categorical theories, exploring conditions under which these independence relations are symmetric and characterizing rosy and superrosy theories.

Contribution

It defines n-ranks and n-independence in countably categorical theories, establishing criteria for symmetry and rosiness based on algebraic closure and exchange properties.

Findings

n-independence is almost an independence relation, possibly lacking symmetry.

Symmetry of n-independence for all n implies the theory is rosy.

Conditions involving algebraic closure and exchange property lead to superrosiness with finite U-thorn-rank.

Abstract

For an $\omega$-categorical theory $T$ and model $\mathcal{M}$ of $T$ we define a hierarchy of ranks, the $n$-ranks for $n < \omega$ 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 $\emptyset$-definable equivalence relation on $n$-tuples of elements from $M$. Using the $n$-rank we define the notion of $n$-independence. For all $n < \omega$, the $n$-independence relation restricted to $M_n$ 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 $n < \omega$, if $\mathcal{M} \models T$ and the algebraic closure in $\mathcal{M}^{\mathrm{eq}}$ 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 $M_n$. Then we show that if $n$-independence is symmetric for all $n < \omega$, then $T$ is rosy. An application of this is that if $T$ has weak elimination of imaginaries and the algebraic closure in $\mathcal{M}$ restricted to elements of $M$ of 0-rank 1 (over some set of parameters from $M^{\mathrm{eq}}$) satisfies the exchange property, then $T$ is superrosy with finite U-thorn-rank.