We develop a new notion of independence (þ-independence, read "thorn"-independence) that arises from a family of ranks suggested by Scanlon (þ-ranks). We prove that in a large class of theories (including simple theories and o-minimal theories) this notion has many of the properties needed for an adequate geometric structure. We prove that þ-independence agrees with the usual independence notions in stable, supersimple and o-minimal theories. Furthermore, we give some evidence that the equivalence between forking and þ-forking in simple theories might (...) be closely related to one of the main open conjectures in simplicity theory, the stable forking conjecture. In particular, we prove that in any simple theory where the stable forking conjecture holds. þ-independence and forking independence agree. (shrink)

Les corps algébriquement clos, réels clos et pseudo-finis n'ont, pour chaque entier n, qu'un nombre fini d'extensions de degré n; nous montrons qu'ils partagent cette propriété avec tous les corps qui, comme eux, satisfont une propriété très rudimentaire de préservation de la dimension, de nature modèle-théorique. Ce résultat est atteint en montrant qu'une certaine action du groupe GLn d'un tel corps n'a qu'un nombre fini d'orbites. /// La korpoj algebre fermataj, reale fermataj kaj pseudofinataj ne havas, pri ciu integro n, (...) pli ol finitan nombron de sternejoj kun grado n; ni montras ke ili kunpartigas tiu ci propecon kun ciu tiel korpoj kiuj kontentigas kiel ili unu tre simplan modelteorian propecon de konserveco de la amplekso. Pri atingi la resultaton, ni montras ke kelka ago de la grupo GLn de tiel korpo havas sole finitan nombron de orbitoj. (shrink)

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.