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Automorphism groups over a hyperimaginary
[article]

2021
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arXiv
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pre-print

Namely we investigate the

arXiv:2106.09200v1
fatcat:b2l3twzqzngwnie2hz57befi2u
*Kim*-Pillay group and the Shelah group, and corresponding notions of*Kim*-Pillay types and Shelah strong types, over a hyperimaginary. ... We now define and characterize the KP(*Kim*-Pillay)-type over a hyperimaginary. Proposition 3.5. ...##
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Around stable forking

2001
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Fundamenta Mathematicae
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These issues were raised in discussions between Hart,

doi:10.4064/fm170-1-6
fatcat:4xqhe3stkfbvzoqkiwi4qrt6za
*Kim*and Pillay in the Fields Institute in the autumn of 1996, but it is quite likely that others have also formulated such problems. ...##
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Independence over arbitrary sets in NSOP_1 theories
[article]

2019
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arXiv
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pre-print

We study

arXiv:1909.08368v1
fatcat:xm7p4gmx6bg4peappgezmjakyy
*Kim*-independence over arbitrary sets. Assuming that forking satisfies existence, we establish Kim's lemma for*Kim*-dividing over arbitrary sets in an NSOP_1 theory. ... We deduce symmetry of*Kim*-independence and the independence theorem for Lascar strong types. ... From the Kim's lemma for*Kim*-dividing we conclude: Proposition 4.1. (*Kim*-forking =*Kim*-dividing) For any A, if ϕ(x; b)*Kim*-forks over A then ϕ(x; b)*Kim*-divides over A. Proof. ...##
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More on tree properties

2019
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Fundamenta Mathematicae
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In addition, we study relationships between TP2 and

doi:10.4064/fm757-8-2019
fatcat:7jketz2fhbck3glc3xyxxrmiqi
*Kim*-forking, and show that a theory is supersimple iff there is no countably infinite*Kim*-forking chain. ... 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 ... A formula*Kim*-forks over A if the formula implies a finite disjunction of formulas, each of which*Kim*-divides over A. (4) A type*Kim*-divides/forks over A if the type implies a formula which*Kim*-divides ...##
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Transitivity, lowness, and ranks in NSOP_1 theories
[article]

2020
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arXiv
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pre-print

We show that, in such theories,

arXiv:2006.10486v1
fatcat:jzpsjerls5apfcwceyyd37jkgy
*Kim*-independence is transitive and that ^K-Morley sequences witness*Kim*-dividing. ... We develop the theory of*Kim*-independence in the context of NSOP_1 theories satsifying the existence axiom. ... is inconsistent. (2) A formula ϕ(x; a) is said to*Kim*-fork over A if ϕ(x; a) ⊢ i<k ψ i (x; c i ), where each ψ i (x; c i )*Kim*-divides over A.(3)We say a type*Kim*-divides (*Kim*-forks) over A if it implies ...##
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Type-amalgamation properties and polygroupoids in stable theories
[article]

2014
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arXiv
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pre-print

We show that in a stable first-order theory, the failure of higher-dimensional type amalgamation can always be witnessed by algebraic structures which we call n-ary polygroupoids. This generalizes a result of Hrushovski that failures of 4-amalgamation in stable theories are witnessed by definable groupoids (which are 2-ary polygroupoids in our terminology). The n-ary polygroupoids are definable in a mild expansion of the language (adding a unary predicate for an infinite Morley sequence).

arXiv:1404.1525v1
fatcat:hm3jhlwixbcg3p3tjagwwlr34y
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Amalgamation functors and homology groups in model theory
[article]

2011
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arXiv
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pre-print

We present definitions of homology groups associated to a family of amalgamation functors. We show that if the generalized amalgamation properties hold, then the homology groups are trivial. We compute the group H_2 for strong types in stable theories and show that in this context, the class of possible groups H_2 is precisely the profinite abelian groups.

arXiv:1105.2921v1
fatcat:s3mhtpk7qnacppv5lwwhknwxzm
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A note on Lascar strong types in simple theories
[article]

1996
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arXiv
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pre-print

Let T be a countable, small simple theory. In this paper, we prove for such T, the notion of Lascar Strong type coincides with the notion of a strong type,over an arbitrary set.

arXiv:math/9608216v1
fatcat:x5bts3xzszgyfj4qras735br3q
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A classification of 2-chains having 1-shell boundaries in rosy theories
[article]

2015
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arXiv
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pre-print

We thank Hyeung-Joon

arXiv:1503.04564v1
fatcat:k46r7whptfeudbcoilljeetpdm
*Kim*and John Goodrick for their valuable suggestions and comments. Notation. Throughout the paper, s denotes an arbitrary finite set of natural numbers. ...##
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The relativized Lascar groups, type-amalgamation, and algebraicity
[article]

2020
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arXiv
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pre-print

Sang-hyun

arXiv:2004.11309v1
fatcat:y4v63g6sobddnhundoijha6ija
*Kim*from Seoul National University. In this paper, we change terminology ". . . -type" to ". . . ... We say a, b have the same KP(*Kim*-Pillay)-type (write a ≡ KP b) if they are in the same class of any bounded ∅-type-definable equivalence relation. • The group Gal L (T ) := Aut(M)/ Autf(M) is called the ...##
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A number of countable models of a countable supersimple theory
[article]

1996
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arXiv
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pre-print

In this paper, we prove the number of countable models of a countable supersimple theory is either 1 or infinite. This result is an extension of Lachlan's theorem on a superstable theory.

arXiv:math/9602217v1
fatcat:ovzyaclchjcwjbwnkmtu4n4lu4
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GEOMETRIC SIMPLICITY THEORY

2009
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Proceedings of the 10th Asian Logic Conference
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Constructing the hyperdefinable group from the group configuration
[article]

2005
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arXiv
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pre-print

For simple theories with a strong version of amalgamation we obtain the canonical hyperdefinable group from the group configuration. This provides a generalization to simple theories of the group configuration theorem for stable theories.

arXiv:math/0508583v1
fatcat:xkmrfgniczehbph5tjpvy3fa4q
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Notions around tree property 1

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

doi:10.1016/j.apal.2011.02.001
fatcat:lcjdspmu4jg4tdaywxqmk4mo2u
## more »

... 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##
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More on tree properties
[article]

2019
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arXiv
*
pre-print

In addition, we study relationships between TP_2 and

arXiv:1902.08911v3
fatcat:hhu44653rzf4ras6ewldrcnwke
*Kim*-forking, and obtain that a theory is supersimple iff there is no countably infinite*Kim*-forking chain. ... Recently it is proved that in any NSOP_1 theory (i.e. a theory not having SOP_1) holding nonforking existence,*Kim*-forking also satisfies all the mentioned independence properties except base monotonicity ... ⌣ | K A B if tp(a/AB) does not*Kim*-fork over A. ...
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