2018/01/17 by Vera Koponen, Koponen, Vera
Decision Sciences · Mathematics · #03C10 #03C35 #03C45 #03C68 #Advanced Topology and Set Theory #FOS: Mathematics #Fuzzy and Soft Set Theory #Logic (math.LO) #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.1801.05748
openalex publication_date 2018/01/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We prove that if T is an ω-categorical supersimple theory with nontrivial dependence (given by forking), then there is a nontrivial regular 1-type over a finite set of reals which is realized by real elements; hence forking induces a nontrivial pregeometry on the solution set of this type and the pregeometry is definable (using only finitely many parameters). The assumption about ω-categoricity is necessary. This result is used to prove the following: If V is a finite relational vocabulary with maximal arity 3 and T is a supersimple V-theory with elimination of quantifiers, then T has trivial dependence and finite SU-rank. This immediately gives the following strengthening of a previous result of the author: if M is a ternary simple homogeneous structure with only finitely many constraints, then Th(M) has trivial dependence and finite SU-rank.