Characterizing rosy theories

Abstract

We examine several conditions, either the existence of a rank or a particular property of þ-forking that suggest the existence of a well-behaved independence relation, and determine the consequences of each of these conditions towards the rosiness of the theory. In particular we show that the existence of an ordinal valued equivalence relation rank is a (necessary and) sufficient condition for rosiness.

Thorn independence in the field of real numbers with a small multiplicative group

“…For instance, the theory of algebraically closed valued fields is not a rosy theory, but þ-forking, restricted to the field, residue field, and value group sorts, is an independence relation. Thus ACVF is real-rosy [4].…”

“…In analogy with simple and stable theories, we make the following definition (which could be equivalently stated in terms of U þ -rank, see [4]):…”

“…To do this, we will need to use the following propositions from [4]. Throughout this section, we assume that (R, G) is κ-saturated, for κ > 2 |L G | Proposition 41.…”

Weakly one-based geometric theories

“…In [13] Gagelman showed that the geometric theories T where the notion of independence extends to the set of imaginary elements are those that are surgical. Recall that a geometric theory T is surgical if whenever X ⊂ M n is denable and dim(X) = m then there is no denable equivalence relation E on X that has innitely many classes of dimension m. The results from [13] together with the fact that thorn forking is the weakest notion of independence [12], show that T is surgical if and only if T is rosy of thorn rank one.…”

Stable embeddedness and NIP

“…There is an extensive theory in place around rosy theories and thorn-forking. See [8] where this was initiated, as well as [5], [2], and [4]. So we can define a theory to be rosy if if every finitary complete type p(x) ∈ S(B) does not thorn-fork over some A ⊆ B of cardinality at most |T |, Rather than define thorn forking, we will give an equivalent definition, referring the reader to the above-mentioned papers for further details and discussions.…”