2023/06/14 by Hanson, James E. · 2 citations
#03C45 #FOS: Mathematics #Logic (math.LO)
paper · doi:10.48550/arxiv.2306.08239
We introduce and examine some special classes of invariant types\unicodex2014bi-invariant, strongly bi-invariant, extendibly invariant, and reliably invariant types\unicodex2014and show that they are related to certain model-theoretic tree properties. We show that the comb tree property (recently introduced by Mutchnik) is equivalent to the failure of Kim's lemma for bi-invariant types and is implied by the failure of Kim's lemma for reliably invariant types over invariance bases. We show that every type over an invariance base extends to a reliably invariant type\unicodex2014generalizing an unpublished result of Kruckman and Ramsey\unicodex2014and use this to show that, under a reasonable definition of Kim-dividing, Kim-forking coincides with Kim-dividing over invariance bases in theories without the comb tree property. Assuming a measurable cardinal, we characterize the comb tree property in terms of a form of dual local character. We also show that the antichain tree property (introduced by Ahn and Kim) seems to have a somewhat similar relationship to strong bi-invariance. In particular, we show that NATP theories satisfy Kim's lemma for strongly bi-invariant types and (assuming a measurable cardinal) satisfy a different form of dual local character. Furthermore, we examine a mutual generalization of the local character properties satisfied by NTP2 and NSOP1 theories and show that it is satisfied by all NATP theories. Finally, we give some related minor results\unicodex2014a strengthened local character characterization of NSOP1 and a characterization of coheirs in terms of invariant extensions in expansions\unicodex2014as well as a pathological example of Kim-dividing.