Notions of rank and independence in countably categorical theories
Vera Koponen
TL;DR
This work develops a hierarchical framework of ranks and independence notions for $\omega$-categorical theories by defining $n$-ranks $\mathrm{rk}^n$ via the algebraic closure restricted to imaginaries up to level $n$, and introducing $n$-independence $\underset{C}{\smile^n}$. It identifies an exchange-property on rank-1 elements that implies symmetry of $\underset{C}{\smile^n}$ on $M_n$, thereby producing genuine independence relations and connecting symmetry across all levels to rosiness. The paper further shows that if the exchange property holds for all $n$ and if imaginaries satisfy soft elimination of imaginaries, then thorn-independence has local character, making $T$ rosy, and under additional hypotheses, superrosy with finite $U^{\thorn}$-rank; these results illuminate a potential dividing line between rosy and non-rosy $\omega$-categorical theories and have implications for Fraïssé limits and related structures.
Abstract
For an $ω$-categorical theory $T$ and model $\mathcal{M}$ 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 $\emptyset$-definable equivalence relation on $n$-tuples of elements from $M$. Using the $n$-rank we define the notion of $n$-independence. For all $n < ω$, 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 $n < ω$, 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 $n < ω$, 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.