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Automorphism groups over a hyperimaginary [article]

Byunghan Kim, Hyoyoon Lee
2021 arXiv   pre-print
Namely we investigate the 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.  ... 
arXiv:2106.09200v1 fatcat:b2l3twzqzngwnie2hz57befi2u

Around stable forking

Byunghan Kim, A. Pillay
2001 Fundamenta Mathematicae  
These issues were raised in discussions between Hart, Kim and Pillay in the Fields Institute in the autumn of 1996, but it is quite likely that others have also formulated such problems.  ... 
doi:10.4064/fm170-1-6 fatcat:4xqhe3stkfbvzoqkiwi4qrt6za

Independence over arbitrary sets in NSOP_1 theories [article]

Jan Dobrowolski, Byunghan Kim, Nicholas Ramsey
2019 arXiv   pre-print
We study 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.  ... 
arXiv:1909.08368v1 fatcat:xm7p4gmx6bg4peappgezmjakyy

More on tree properties

Enrique Casanovas, Byunghan Kim
2019 Fundamenta Mathematicae  
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.  ...  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  ... 
doi:10.4064/fm757-8-2019 fatcat:7jketz2fhbck3glc3xyxxrmiqi

Transitivity, lowness, and ranks in NSOP_1 theories [article]

Artem Chernikov, Byunghan Kim, Nicholas Ramsey
2020 arXiv   pre-print
We show that, in such theories, 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  ... 
arXiv:2006.10486v1 fatcat:jzpsjerls5apfcwceyyd37jkgy

Type-amalgamation properties and polygroupoids in stable theories [article]

John Goodrick, Byunghan Kim, Alexei Kolesnikov
2014 arXiv   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

Amalgamation functors and homology groups in model theory [article]

John Goodrick, Byunghan Kim, Alexei Kolesnikov
2011 arXiv   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

A note on Lascar strong types in simple theories [article]

Byunghan Kim
1996 arXiv   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

A classification of 2-chains having 1-shell boundaries in rosy theories [article]

Byunghan Kim, SunYoung Kim, Junguk Lee
2015 arXiv   pre-print
We thank Hyeung-Joon Kim and John Goodrick for their valuable suggestions and comments. Notation. Throughout the paper, s denotes an arbitrary finite set of natural numbers.  ... 
arXiv:1503.04564v1 fatcat:k46r7whptfeudbcoilljeetpdm

The relativized Lascar groups, type-amalgamation, and algebraicity [article]

Jan Dobrowolski, Byunghan Kim, Alexei Kolesnikov, Junguk Lee
2020 arXiv   pre-print
Sang-hyun 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  ... 
arXiv:2004.11309v1 fatcat:y4v63g6sobddnhundoijha6ija

A number of countable models of a countable supersimple theory [article]

Byunghan Kim
1996 arXiv   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

GEOMETRIC SIMPLICITY THEORY

BYUNGHAN KIM
2009 Proceedings of the 10th Asian Logic Conference  
doi:10.1142/9789814293020_0009 fatcat:ezwxwxhptfh5lfzfc7tohynmee

Constructing the hyperdefinable group from the group configuration [article]

Tristram De Piro, Byunghan Kim, Jessica Millar
2005 arXiv   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

Notions around tree property 1

Byunghan Kim, Hyeung-Joon Kim
2011 Annals of Pure and Applied Logic  
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
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
doi:10.1016/j.apal.2011.02.001 fatcat:lcjdspmu4jg4tdaywxqmk4mo2u

More on tree properties [article]

Enrique Casanovas, Byunghan Kim
2019 arXiv   pre-print
In addition, we study relationships between TP_2 and 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.  ... 
arXiv:1902.08911v3 fatcat:hhu44653rzf4ras6ewldrcnwke
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