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57
Sep 22, 2013
09/13
by
Byunghan Kim
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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.
Source: http://arxiv.org/abs/math/9602217v1
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69
Sep 23, 2013
09/13
by
Byunghan Kim
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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.
Source: http://arxiv.org/abs/math/9608216v1
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54
Sep 23, 2013
09/13
by
John Goodrick; Byunghan Kim; Alexei Kolesnikov
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This paper continues the study of generalized amalgamation properties. Part of the paper provides a finer analysis of the groupoids that arise from failure of 3-uniqueness in a stable theory. We show that such groupoids must be abelian and link the binding group of the groupoids to a certain automorphism group of the monster model, showing that the group must be abelian as well. We also study connections between n-existence and n-uniqueness properties for various "dimensions" n in the...
Source: http://arxiv.org/abs/1006.4410v2
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6.0
Jun 30, 2018
06/18
by
John Goodrick; Byunghan Kim; Alexei Kolesnikov
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We give an explicit description of the homomorphism group H_n(p) of a strong type p in any stable theory under the assumption that for every non-forking extension q of p the groups H_i(q) are trivial for i at least 2 but less than n. The group H_n(p) turns out to be isomorphic to the automorphism group of a certain piece of the algebraic closure of n independent realizations of p; it was shown earlier by the authors that such a group must be abelian. We call this the "Hurewicz...
Topics: Mathematics, Logic
Source: http://arxiv.org/abs/1412.3864
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15
Jun 27, 2018
06/18
by
Byunghan Kim; SunYoung Kim; Junguk Lee
texts
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We construct a possibly non-commutative groupoid from the failure of $3$-uniqueness of a strong type. The commutative groupoid constructed by John Goodrick and Alexei Kolesnikov in \cite{GK} lives in the center of the groupoid. A certain automorphism group approximated by the vertex groups of the non-commutative groupoids is suggested as a "fundamental group" of the strong type.
Topics: Logic, Mathematics
Source: http://arxiv.org/abs/1504.07731
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3.0
Jun 30, 2018
06/18
by
John Goodrick; Byunghan Kim; Alexei Kolesnikov
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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).
Topics: Mathematics, Logic
Source: http://arxiv.org/abs/1404.1525
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45
Sep 23, 2013
09/13
by
John Goodrick; Byunghan Kim; Alexei Kolesnikov
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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.
Source: http://arxiv.org/abs/1105.2921v1
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16
Jun 27, 2018
06/18
by
Byunghan Kim; SunYoung Kim; Junguk Lee
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We classify, in a non-trivial amenable collection of functors, all 2-chains up to the relation of having the same 1-shell boundary. In particular, we prove that in a rosy theory, every 1-shell of a Lascar strong type is the boundary of some 2-chain, hence making the 1st homology group trivial. We also show that, unlike in simple theories, in rosy theories there is no upper bound on the minimal lengths of $2$-chains whose boundary is a $1$-shell.
Topics: Logic, Mathematics
Source: http://arxiv.org/abs/1503.04564
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56
Sep 21, 2013
09/13
by
Tristram De Piro; Byunghan Kim; Jessica Millar
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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.
Source: http://arxiv.org/abs/math/0508583v1
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Sep 23, 2013
09/13
by
Byunghan Kim; Hyeung-Joon Kim; Lynn Scow
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We give definitions that distinguish between two notions of indiscernibility for a set $\{a_\eta \mid \eta \in \W\}$ that saw original use in \cite{sh90}, which we name \textit{$\s$-} and \textit{$\n$-indiscernibility}. Using these definitions and detailed proofs, we prove $\s$- and $\n$-modeling theorems and give applications of these theorems. In particular, we verify a step in the argument that TP is equivalent to TP$_1$ or TP$_2$ that has not seen explication in the literature. In the...
Source: http://arxiv.org/abs/1111.0915v2
