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Notions around tree property 1

Byunghan Kim, Hyeung-Joon Kim

2011
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Annals of Pure and Applied Logic
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In this paper, we study the notions related to tree property 1 (=TP 1 ), or, equivalently, SOP 2 . Among others, we supply a type-counting criterion for TP 1 and show the equivalence of TP 1 and k-TP 1 . Then we introduce the notions of weak k-TP 1 for k ≥ 2, and also supply typecounting criteria for those. We do not know whether weak k-TP 1 implies TP 1 , but at least we prove that each weak k-TP 1 implies SOP 1 . Our generalization of the tree-indiscernibility results in Džamonja and Shelah
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... 004) [5] is crucially used throughout the paper. As is well known, a complete theory T is simple if and only if it does not have the tree property. A theory being simple is characterized by its having an (automorphism-invariant) independence relation satisfying symmetry, transitivity, extension (i.e., for any c and A ⊆ B, there is c ′ (≡ A c) such that c ′ is independent with B over A), local character, finite character, antireflexivity (a tuple c is always dependent with itself over any set B unless c ∈ acl(B)), and type amalgamation over a model [10] . But still, it is natural to ask whether there is a suitable class of theories (possibly properly containing that of simple theories) having an independence relation satisfying a smaller number of the aforementioned independence axioms. Indeed the class of rosy theories is characterized by having an independence relation for M eq satisfying all the axioms except for type amalgamation over a model. Thus, all simple and o-minimal theories are rosy [6,1]. On the other hand, there are natural examples (which need not be rosy) having an independence relation for M eq satisfying all the aforementioned axioms including stationarity over a model (which implies type amalgamation over a model), except for local character. In [2], such theories are called mock stable or mock simple, respectively. Example 0.1. (1) (The random parameterized equivalence relations.) Let T 0 be a theory with two sorts (P, E) and a ternary relation ∼ on P × P × E saying that, for each e ∈ E, x ∼ e y forms an equivalence relation on P. Let T be a Fraïssé limit theory of the class of finite models of T 0 . For sets A, B, C ⊆ M eq (|= T eq ), we put A ⌣ | C B iff acl(ACE) ∩ acl(BCE) = acl(CE) in M eq , where E indeed means E(M). One can easily check that ⌣ | witnesses mock stability of T , but T is not rosy and, in particular, not simple (see [1, 1.7, 1.15, 1.55]). The failure of local character for ⌣ | is witnessed by {e i ∈ E| i ∈ κ} and p ∈ P with c i = p/e i , as we have {c j e j | j<i} c i e i for each i < κ. (2) (A vector space with a bilinear form.) In [7] , Granger supplied a model theory of bilinear forms. In particular, he studied two sorted structure (V , K ), where V is a vector space over an algebraically closed field K (of some fixed characteristic) with a nondegenerate reflexive bilinear form. Any two such structures (V 1 , K 1 ) and (V 2 , K 2 ) are elementarily equivalent

doi:10.1016/j.apal.2011.02.001
fatcat:lcjdspmu4jg4tdaywxqmk4mo2u