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.
"Properties and consequences of Thorn-independence." J. Symbolic Logic 71 (1) 1 - 21, March 2006. https://doi.org/10.2178/jsl/1140641160