2022/02/13 by Santiago Cárdenas-Martín, Cárdenas-Martín, Santiago, Rafel Farré +1
Mathematics · #Advanced Topology and Set Theory #FOS: Mathematics #Limits and Structures in Graph Theory #Logic (math.LO) #Markov Chains and Monte Carlo Methods
paper · pdf · doi:10.48550/arxiv.2202.06395
openalex publication_date 2022/02/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Just as Lascar's notion of abstract rank axiomatizes the U-rank, we propose axioms for the ranks SUd and SUf, the foundation ranks of dividing and forking. We study the relationships between these axioms. As with superstable, we characterize supersimple types and theories based on the existence of these ranks. We show that the U-rank is the foundation rank of the Lascar-splitting independence relationship. We also provide an alternative definition of SUd similar to the original definition of U. Finally, we check that if in the standard characterizations of simple and supersimple we change the non-forking independence for the non-lascar-splitting independence, we characterize stable and superstable.