Tree properties were introduced by Shelah, and it is well-known that a theory has TP (the tree property) if and only if it has TP1 or TP2. In any simple theory (i.e., a theory not having TP), forking supplies a good independence notion as it satisfies symmetry, full transitivity, extension, local character, and type-amalgamation, over sets. Shelah also introduced SOPn (n-strong order property). Recently it has been proved that in any NSOP1 theory (i.e. a theory not having SOP1) having nonforking existence, Kim-forking also satisfies all the above mentioned independence properties except base monotonicity (one direction of full transitivity). These results are the sources of motivation for this paper. Mainly, we produce type-counting criteria for SOP2 (which is equivalent to TP1) and SOP1. In addition, we study relationships between TP2 and Kim-forking, and show that a theory is supersimple iff there is no countably infinite Kim-forking chain.
Bibliographical noteFunding Information:
The first author has been partially funded by a Spanish government grant MTM2017-86777-P and a Catalan DURSI grant 2017SGR-270. The second author has been supported by Samsung Science Technology Foundation under Project Number SSTF-BA1301-03 and an NRF of Korea grant 2018R1D1A1A02085584.
Acknowledgments. The first author has been partially funded by a
All Science Journal Classification (ASJC) codes
- Algebra and Number Theory