Byunghan Kim

Search for another author

Author Description

Ask a Question About This Mathematician

All published works (60)

Existence in NSOP$_1$ theories

2024-05-22

We show that Kim-forking satisfies existence in all NSOP$_1$ theories. We show that Kim-forking satisfies existence in all NSOP$_1$ theories.

Chat View PDF

TRANSITIVITY, LOWNESS, AND RANKS IN NSOP THEORIES

2023-06-09

Artem Chernikov , Byunghan Kim , Nicholas Ramsey

Abstract We develop the theory of Kim-independence in the context of NSOP $_{1}$ theories satisfying the existence axiom. We show that, in such theories, Kim-independence is transitive and that -Morley … Abstract We develop the theory of Kim-independence in the context of NSOP $_{1}$ theories satisfying the existence axiom. We show that, in such theories, Kim-independence is transitive and that -Morley sequences witness Kim-dividing. As applications, we show that, under the assumption of existence, in a low NSOP $_{1}$ theory, Shelah strong types and Lascar strong types coincide and, additionally, we introduce a notion of rank for NSOP $_{1}$ theories.

Chat View PDF

Automorphism groups over a hyperimaginary

2022-06-17

Byunghan Kim , Hyoyoon Lee

In this paper we study the Lascar group over a hyperimaginary $\boldsymbol{e}$. We verify that various results about the group over a real set still hold when the set is … In this paper we study the Lascar group over a hyperimaginary $\boldsymbol{e}$. We verify that various results about the group over a real set still hold when the set is replaced by $\boldsymbol{e}$. First of all, there is no written proof in the available literature that the group over $\boldsymbol{e}$ is a topological group. We present an expository style proof of the fact, which even simplifies existing proofs for the real case. We further extend a result that the orbit equivalence relation under a closed subgroup of the Lascar group is type-definable. On the one hand, we correct errors appeared in the book written by the first author and produce a counterexample. On the other hand, we extend Newelski's theorem that ‘a G-compact theory over a set has a uniform bound for the Lascar distances’ to the hyperimaginary context. Lastly, we supply a partial positive answer to a question about the kernel of a canonical projection between relativized Lascar groups, which is even a new result in the real context.

Chat View PDF

Independence over arbitrary sets in NSOP1 theories

2021-10-25

Jan Dobrowolski , Byunghan Kim , Nicholas Ramsey

Chat View PDF

WEAK CANONICAL BASES IN NSOP THEORIES

2021-06-11

Byunghan Kim

Abstract We study the notion of weak canonical bases in an NSOP $_{1}$ theory T with existence. Given $p(x)=\operatorname {tp}(c/B)$ where $B=\operatorname {acl}(B)$ in ${\mathcal M}^{\operatorname {eq}}\models T^{\operatorname {eq}}$ , … Abstract We study the notion of weak canonical bases in an NSOP $_{1}$ theory T with existence. Given $p(x)=\operatorname {tp}(c/B)$ where $B=\operatorname {acl}(B)$ in ${\mathcal M}^{\operatorname {eq}}\models T^{\operatorname {eq}}$ , the weak canonical base of p is the smallest algebraically closed subset of B over which p does not Kim-fork. With this aim we firstly show that the transitive closure $\approx $ of collinearity of an indiscernible sequence is type-definable. Secondly, we prove that given a total $\mathop {\smile \hskip -0.9em ^| \ }^K$ -Morley sequence I in p , the weak canonical base of $\operatorname {tp}(I/B)$ is $\operatorname {acl}(e)$ , if the hyperimaginary $I/\approx $ is eliminable to e , a sequence of imaginaries. We also supply a couple of criteria for when the weak canonical base of p exists. In particular the weak canonical base of p is (if exists) the intersection of the weak canonical bases of all total $\mathop {\smile \hskip -0.9em ^| \ }^K$ -Morley sequences in p over B . However, while we investigate some examples, we point out that given two weak canonical bases of total $\mathop {\smile \hskip -0.9em ^| \ }^K$ -Morley sequences in p need not be interalgebraic, contrary to the case of simple theories. Lastly we suggest an independence relation relying on weak canonical bases, when T has those. The relation, satisfying transitivity and base monotonicity, might be useful in further studies on NSOP $_1$ theories .

THE RELATIVIZED LASCAR GROUPS, TYPE-AMALGAMATION, AND ALGEBRAICITY

2021-05-06

Jan Dobrowolski , Byunghan Kim , Alexei Kolesnikov +1 more

Abstract In this paper we study the relativized Lascar Galois group of a strong type. The group is a quasi-compact connected topological group, and if in addition the underlying theory … Abstract In this paper we study the relativized Lascar Galois group of a strong type. The group is a quasi-compact connected topological group, and if in addition the underlying theory T is G -compact, then the group is compact. We apply compact group theory to obtain model theoretic results in this note. For example, we use the divisibility of the Lascar group of a strong type to show that, in a simple theory, such types have a certain model theoretic property that we call divisible amalgamation. The main result of this paper is that if c is a finite tuple algebraic over a tuple a , the Lascar group of $\operatorname {stp}(ac)$ is abelian, and the underlying theory is G -compact, then the Lascar groups of $\operatorname {stp}(ac)$ and of $\operatorname {stp}(a)$ are isomorphic. To show this, we prove a purely compact group-theoretic result that any compact connected abelian group is isomorphic to its quotient by every finite subgroup. Several (counter)examples arising in connection with the theoretical development of this note are presented as well. For example, we show that, in the main result above, neither the assumption that the Lascar group of $\operatorname {stp}(ac)$ is abelian, nor the assumption of c being finite can be removed.

Chat View PDF

Automorphism groups over a hyperimaginary

2021-01-01

Byunghan Kim , Hyoyoon Lee

In this paper we study the Lascar group over a hyperimaginary e. We verify that various results about the group over a real set still hold when the set is … In this paper we study the Lascar group over a hyperimaginary e. We verify that various results about the group over a real set still hold when the set is replaced by e. First of all, there is no written proof in the available literature that the group over e is a topological group. We present an expository style proof of the fact, which even simplifies existing proofs for the real case. We further extend a result that the orbit equivalence relation under a closed subgroup of the Lascar group is type-definable. On the one hand, we correct errors appeared in the book, "Simplicity Theory" [6, 5.1.14-15] and produce a counterexample. On the other, we extend Newelski's Theorem in "The diameter of a Lascar strong type" [12] that `a G-compact theory over a set has a uniform bound for the Lascar distances' to the hyperimaginary context. Lastly, we supply a partial positive answer to a question raised in "The relativized Lascar groups, type-amalgamations, and algebraicity" [4, 2.11], which is even a new result in the real context.

Chat View PDF

The relativized Lascar groups, type-amalgamation, and algebraicity

2020-04-23

Jan Dobrowolski , Byunghan Kim , Alexei Kolesnikov +1 more

We apply compact group theory to obtain some model-theoretic results about the relativized Lascar Galois group of a strong type. We apply compact group theory to obtain some model-theoretic results about the relativized Lascar Galois group of a strong type.

Chat View PDF

Transitivity, lowness, and ranks in NSOP$_1$ theories

2020-01-01

Artem Chernikov , Byunghan Kim , Nicholas Ramsey

We develop the theory of Kim-independence in the context of NSOP$_{1}$ theories satsifying the existence axiom. We show that, in such theories, Kim-independence is transitive and that $\ind^{K}$-Morley sequences witness … We develop the theory of Kim-independence in the context of NSOP$_{1}$ theories satsifying the existence axiom. We show that, in such theories, Kim-independence is transitive and that $\ind^{K}$-Morley sequences witness Kim-dividing. As applications, we show that, under the assumption of existence, in a low NSOP$_{1}$ theory, Shelah strong types and Lascar strong types coincide and, additionally, we introduce a notion of rank for NSOP$_{1}$ theories.

Chat View PDF

The relativized Lascar groups, type-amalgamation, and algebraicity

2020-01-01

Jan Dobrowolski , Byunghan Kim , Alexei Kolesnikov +1 more

Chat View PDF

More on tree properties

2019-12-19

Enrique Casanovas , Byunghan Kim

Tree properties were introduced by Shelah, and it is well-known that a theory has TP (the tree property) if and only if it has ${\rm TP}_1$ or ${\rm TP}_2$. In … Tree properties were introduced by Shelah, and it is well-known that a theory has TP (the tree property) if and only if it has ${\rm TP}_1$ or ${\rm TP}_2$. In any simple theory (i.e., a theory not having TP), forking supplies a good indepen

Non-commutative groupoids obtained from the failure of 3-uniqueness in stable theories

2019-11-25

Byunghan Kim , Sunyoung Kim , Junguk Lee

Given an arbitrary connected groupoid $\mathcal {G}$ with its vertex group ${\mathcal G} _a$, if ${\mathcal G} _a$ is a central subgroup of a group $F$, then there is a … Given an arbitrary connected groupoid $\mathcal {G}$ with its vertex group ${\mathcal G} _a$, if ${\mathcal G} _a$ is a central subgroup of a group $F$, then there is a canonical extension $\mathcal {F}=\mathcal {G}\otimes F$ of ${\mathcal G} $ in the

Chat View PDF

Independence over arbitrary sets in NSOP$_1$ theories

2019-09-18

Jan Dobrowolski , Byunghan Kim , Nicholas Ramsey

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 … 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.

Chat View PDF

ON THE NUMBER OF COUNTABLE MODELS OF A COUNTABLE NSOP1 THEORY WITHOUT WEIGHT ω

2019-07-02

Byunghan Kim

Abstract In this article, we prove that if a countable non- ${\aleph _0}$ -categorical NSOP 1 theory with nonforking existence has finitely many countable models, then there is a finite … Abstract In this article, we prove that if a countable non- ${\aleph _0}$ -categorical NSOP 1 theory with nonforking existence has finitely many countable models, then there is a finite tuple whose own preweight is ω . This result is an extension of a theorem of the author on any supersimple theory.

More on tree properties

2019-02-24

Enrique Casanovas , Byunghan Kim

Tree properties are introduced by Shelah, and it is well-known that a theory has TP (the tree property) if and only if it has TP$_1$ or TP$_2$. In any simple … Tree properties are introduced by Shelah, and it is well-known that a theory has TP (the tree property) if and only if it has TP$_1$ or TP$_2$. In any simple theory (i.e., a theory not having TP), forking supplies a good independence notion as it satisfies symmetry, transitivity, extension, local character, and type-amalgamation. Shelah also introduced SOP$_n$ ($n$-strong order property). 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 (one direction of transitivity). These results are the sources of motivation for this paper. Mainly, we produce type-counting criteria for SOP$_2$ (which is equivalent to TP$_1$) and SOP$_1$. 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.

Chat View PDF

Independence over arbitrary sets in NSOP$_1$ theories

2019-01-01

Jan Dobrowolski , Byunghan Kim , Nicholas Ramsey

Chat View PDF

More on tree properties

2019-01-01

Enrique Casanovas , Byunghan Kim

Chat View PDF

The Lascar groups and the first homology groups in model theory

2017-07-04

Jan Dobrowolski , Byunghan Kim , Junguk Lee

Chat View PDF

Homology groups of types in stable theories and the Hurewicz correspondence

2017-03-18

John Goodrick , Byunghan Kim , Alexei Kolesnikov

Chat View PDF

On the n-uniqueness of types in rosy theories

2016-04-23

Byunghan Kim

The Lascar groups and the 1st homology groups in model theory

2015-11-11

Jan Dobrowolski , Byunghan Kim , Junguk Lee

Let $p$ be a strong type of an algebraically closed tuple over $B=\acl^{\eq}(B)$ in any theory $T$. Depending on a ternary relation $\indo^*$ satisfying some basic axioms (there is at … Let $p$ be a strong type of an algebraically closed tuple over $B=\acl^{\eq}(B)$ in any theory $T$. Depending on a ternary relation $\indo^*$ satisfying some basic axioms (there is at least one such, namely the trivial independence in $T$), the first homology group $H^*_1(p)$ can be introduced, similarly to \cite{GKK1}. We show that there is a canonical surjective homomorphism from the Lascar group over $B$ to $H^*_1(p)$. We also notice that the map factors naturally via a surjection from the `relativised' Lascar group of the type (which we define in analogy with the Lascar group of the theory) onto the homology group, and we give an explicit description of its kernel. Due to this characterization, it follows that the first homology group of $p$ is independent from the choice of $\indo^*$, and can be written simply as $H_1(p)$. As consequences, in any $T$, we show that $|H_1(p)|\geq 2^{\aleph_0}$ unless $H_1(p)$ is trivial, and we give a criterion for the equality of stp and Lstp of algebraically closed tuples using the notions of the first homology group and a relativised Lascar group. We also argue how any abelian connected compact group can appear as the first homology group of the type of a model.

Chat View PDF

Type-amalgamation properties and polygroupoids in stable theories

2015-05-26

John Goodrick , Byunghan Kim , Alexei Kolesnikov

We show that in a stable first-order theory, the failure of higher dimensional type amalgamation can always be witnessed by algebraic structures that we call n-ary polygroupoids. This generalizes a … We show that in a stable first-order theory, the failure of higher dimensional type amalgamation can always be witnessed by algebraic structures that we call n-ary polygroupoids. This generalizes a result of Hrushovski in [16] that failures of 4-amalgamation are witnessed by definable groupoids (which correspond to 2-ary polygroupoids in our terminology). The n-ary polygroupoids are definable in a mild expansion of the language (adding a predicate for a Morley sequence).

Chat View PDF

A CLASSIFICATION OF 2-CHAINS HAVING 1-SHELL BOUNDARIES IN ROSY THEORIES

2015-03-01

Byunghan Kim , Sunyoung Kim , Junguk Lee

Abstract We classify, in a nontrivial 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 … Abstract We classify, in a nontrivial 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.

Chat View PDF

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

2015-01-01

Byunghan Kim , SunYoung Kim , Junguk Lee

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, … 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.

Chat View PDF

Non-commutative groupoids obtained from the failure of $3$-uniqueness in stable theories

2015-01-01

Byunghan Kim , SunYoung Kim , Junguk Lee

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 … 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.

Chat View PDF

Homology groups of types in stable theories and the Hurewicz correspondence

2014-12-12

John Goodrick , Byunghan Kim , Alexei Kolesnikov

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 … 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 correspondence in analogy with the Hurewicz Theorem in algebraic topology.

Chat View PDF

Type-amalgamation properties and polygroupoids in stable theories

2014-04-05

John Goodrick , Byunghan Kim , Alexei Kolesnikov

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 … 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).

Chat View PDF

Type-amalgamation properties and polygroupoids in stable theories

2014-04-05

John Goodrick , Byunghan Kim , Alexei Kolesnikov

Chat View PDF

Type-amalgamation properties and polygroupoids in stable theories

2014-01-01

John Goodrick , Byunghan Kim , Alexei Kolesnikov

Chat View PDF

Homology groups of types in stable theories and the Hurewicz correspondence

2014-01-01

John Goodrick , Byunghan Kim , Alexei Kolesnikov

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 … 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 correspondence" in analogy with the Hurewicz Theorem in algebraic topology.

Chat View PDF

Tree indiscernibilities, revisited

2013-12-19

Byunghan Kim , Hyeungjoon Kim , Lynn Scow

Chat View PDF

Homology Groups of Types in Model Theory and the Computation of H2(p)

2013-12-01

John Goodrick , Byunghan Kim , Alexei Kolesnikov

Abstract We present definitions of homology groups H n ( p ), n ≥ 0, associated to a complete type p . We show that if the generalized amalgamation properties … Abstract We present definitions of homology groups H n ( p ), n ≥ 0, associated to a complete type p . We show that if the generalized amalgamation properties hold, then the homology groups are trivial. We compute the group H 2 ( p ) for strong types in stable theories and show that any profinite abelian group can occur as the group H 2 ( p ).

Elimination of Hyperimaginaries

2013-10-17

Byunghan Kim

This chapter addresses the issues on elimination of hyperimaginaries. The Lascar topological group of automorphisms and its quotient groups are investigated as well. A theory has elimination of hyperimaginaries if … This chapter addresses the issues on elimination of hyperimaginaries. The Lascar topological group of automorphisms and its quotient groups are investigated as well. A theory has elimination of hyperimaginaries if on each complete type, every type-definable equivalence relation is a conjunction of definable equivalence relation. That low (small, resp.) simple theories eliminate bounded (finitary, resp.) hyperimaginaries is shown. The main result by Buechler, Pillay, and Wagner is that any supersimple theory eliminates hyperimaginaries. The result has consequences in regard to the definability and canonical bases of Lascar types.

Lascar Strong Types and Type Amalgamation

2013-10-17

Byunghan Kim

This chapter is devoted to Lascar strong types, and type amalgamation over Lascar strong types is proved. A goal is to show that basic independence relations together with type amalgamation … This chapter is devoted to Lascar strong types, and type amalgamation over Lascar strong types is proved. A goal is to show that basic independence relations together with type amalgamation characterize simplicity and forking.

Generalized Amalgamation and the Group Configuration Theorem

2013-10-17

Byunghan Kim

Abstract This chapter continues from the previous chapter for arbitrary context. The group configuration theorem is established under generalized amalgamation. Then by combining outcomes in Chapter 7, a canonical vector … Abstract This chapter continues from the previous chapter for arbitrary context. The group configuration theorem is established under generalized amalgamation. Then by combining outcomes in Chapter 7, a canonical vector space over the division ring of endogenies is hyperdefinably recovered from anynon-trivial modular structure. Generalized amalgamation notion itself is studied as well.

Groups

2013-10-17

Byunghan Kim

The chapter develops the theory of groups in simple structures, which have beenmostly established by F. O. Wagner following A. Pillay’s initial work.The group can either be type-definable or hyperdefinable. … The chapter develops the theory of groups in simple structures, which have beenmostly established by F. O. Wagner following A. Pillay’s initial work.The group can either be type-definable or hyperdefinable. Commensurativity, a generalization of the group theoretic notion commensurability, and local connectivity play crucial roles in developing the hyperdefinable group theory. Important results on 1-based groups and supersimple groups are stated. In particular, the canonical vector space structure found in a minimal connected 1-based group will be used in Chapter 9, together with the process of obtaining a hyperdefinable group and its hyperdefinable action from generically given data.

The Lascar Group and the Strong Types of Hyperimaginaries

2013-01-01

Byunghan Kim

This is an expository note on the Lascar group. We also study the Lascar group over hyperimaginaries and make some new observations on the strong types over those. In particular, … This is an expository note on the Lascar group. We also study the Lascar group over hyperimaginaries and make some new observations on the strong types over those. In particular, we show that in a simple theory Ltp≡stp in real context implies that for hyperimaginary context.

Amalgamation functors and boundary properties in simple theories

2012-07-02

John Goodrick , Byunghan Kim , Alexei Kolesnikov

Chat View PDF

Notions around tree property 1

2011-03-07

Byunghan Kim , Hyeung-Joon Kim

Amalgamation functors and homology groups in model theory

2011-01-01

John Goodrick , Byunghan Kim , Alexei Kolesnikov

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 … 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.

Chat View PDF

Amalgamation functors and boundary properties in simple theories

2010-06-23

John Goodrick , Byunghan Kim , Alexei Kolesnikov

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. … 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 wider context of simple theories. We introduce a family of weaker existence and uniqueness properties. Many of these properties did appear in the literature before; we give a category-theoretic formulation and study them systematically. Finally, we give examples of first-order simple unstable theories showing, in particular, that there is no straightforward generalization of the groupoid construction in an unstable context.

Chat View PDF

Recovering the hyperdefinable group action in the group configuration theorem

2010-01-25

Byunghan Kim

Abstract In this paper, we continue the construction done in [3], so that under model-4-CA or 4-CA, given a bounded quadrangle C induced from a group configuration, we build a … Abstract In this paper, we continue the construction done in [3], so that under model-4-CA or 4-CA, given a bounded quadrangle C induced from a group configuration, we build a canonical hyperdefinable homogeneous space equivalent to C . When C is principal, we can choose the homogeneous space principal as well.

Amalgamation functors and boundary properties in simple theories

2010-01-01

John Goodrick , Byunghan Kim , Alexei Kolesnikov

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. … 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 wider context of simple theories. We introduce a family of weaker existence and uniqueness properties. Many of these properties did appear in the literature before; we give a category-theoretic formulation and study them systematically. Finally, we give examples of first-order simple unstable theories showing, in particular, that there is no straightforward generalization of the groupoid construction in an unstable context.

Chat View PDF

Generalized amalgamation and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" display="inline" overflow="scroll"><mml:mi>n</mml:mi></mml:math>-simplicity

2008-04-23

Byunghan Kim , Alexei Kolesnikov , Akito Tsuboi

Stable definability and generic relations

2007-12-01

Byunghan Kim , Rahim Moosa

Abstract An amalgamation base p in a simple theory is stably definable if its canonical base is interde-finable with the set of canonical parameters for the ϕ-definitions of p as … Abstract An amalgamation base p in a simple theory is stably definable if its canonical base is interde-finable with the set of canonical parameters for the ϕ-definitions of p as ϕ ranges through all stable formulae. A necessary condition for stably definability is given and used to produce an example of a supersimple theory with stable forking having types that are not stably definable. This answers negatively a question posed in [8]. A criterion for and example of a stably definable amalgamation base whose restriction to the canonical base is not axiomatised by stable formulae are also given. The examples involve generic relations over non CM-trivial stable theories.

Chat View PDF

The group configuration theorem and its applications (モデル理論における独立概念と次元--RIMS研究集会報告集)

2007-05-01

Byunghan Kim

A note on weak dividing

2007-01-04

Byunghan Kim , Niandong Shi

CONSTRUCTING THE HYPERDEFINABLE GROUP FROM THE GROUP CONFIGURATION

2006-12-01

Tristram de Piro , Byunghan Kim , Jessica Millar

Under [Formula: see text]-amalgamation, we obtain the canonical hyperdefinable group from the group configuration. Under [Formula: see text]-amalgamation, we obtain the canonical hyperdefinable group from the group configuration.

Chat View PDF

Constructing the hyperdefinable group from the group configuration

2005-01-01

Tristram de Piro , Byunghan Kim , Jessica Millar

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 … 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.

Chat View PDF

NON-FINITE AXIOMATIZABILITY (Generic structures and their applications)

2004-07-01

Byunghan Kim

Common Coauthors

Coauthor	Papers Together
Alexei Kolesnikov	16
John Goodrick	12
Junguk Lee	9
Jan Dobrowolski	8
Nicholas Ramsey	5
Enrique Casanovas	4
Tristram de Piro	3
Jessica Millar	2
Hyoyoon Lee	2
SunYoung Kim	2
Artem Chernikov	2
Sunyoung Kim	2
Anand Pillay	2
Rahim Moosa	1
Bradd Hart	1
Akito Tsuboi	1
Hyeungjoon Kim	1
H. C. Lee	1
Joonhee Kim	1
Niandong Shi	1
Lynn Scow	1
Hyeung-Joon Kim	1

Commonly Cited References

Homology Groups of Types in Model Theory and the Computation of H2(p)

2013-12-01

John Goodrick , Byunghan Kim , Alexei Kolesnikov

Groupoids, covers, and 3-uniqueness in stable theories

2010-07-09

John Goodrick , Alexei Kolesnikov

Abstract Building on Hrushovski's work in [5], we study definable groupoids in stable theories and their relationship with 3-uniqueness and finite internal covers. We introduce the notion of retractability of … Abstract Building on Hrushovski's work in [5], we study definable groupoids in stable theories and their relationship with 3-uniqueness and finite internal covers. We introduce the notion of retractability of a definable groupoid (which is slightly stronger than Hrushovski's notion of eliminability), give some criteria for when groupoids are retractable, and show how retractability relates to both 3-uniqueness and the splitness of finite internal covers. One application we give is a new direct method of constructing non-eliminable groupoids from witnesses to the failure of 3-uniqueness. Another application is a proof that any finite internal cover of a stable theory with a centerless liaison groupoid is almost split.

GALOIS GROUPS OF FIRST ORDER THEORIES

2001-11-01

Enrique Casanovas , Daniel Lascar , Anthony L. Pillay +1 more

We study the groups Gal L (T) and Gal KP (T), and the associated equivalence relations E L and E KP , attached to a first order theory T. An … We study the groups Gal L (T) and Gal KP (T), and the associated equivalence relations E L and E KP , attached to a first order theory T. An example is given where E L ≠ E KP (a non G-compact theory). It is proved that E KP is the composition of E L and the closure of E L . Other examples are given showing this is best possible.

Characterizing rosy theories

2007-09-01

Clifton Ealy , Alf Onshuus

Abstract We examine several conditions, either the existence of a rank or a particular property of þ-forking that suggest the existence of a well-behaved independence relation, and determine the consequences … Abstract We examine several conditions, either the existence of a rank or a particular property of þ-forking that suggest the existence of a well-behaved independence relation, and determine the consequences of each of these conditions towards the rosiness of the theory. In particular we show that the existence of an ordinal valued equivalence relation rank is a (necessary and) sufficient condition for rosiness.

Amalgamation functors and boundary properties in simple theories

2012-07-02

John Goodrick , Byunghan Kim , Alexei Kolesnikov

Chat View PDF

A note on Lascar strong types in simple theories

1998-09-01

Byunghan Kim

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

Chat View PDF

On the category of models of a complete theory

1982-06-01

Daniel Lascar

Let T be a countable complete theory and C ( T ) the category whose objects are the models of T and morphisms are the elementary maps. The main object … Let T be a countable complete theory and C ( T ) the category whose objects are the models of T and morphisms are the elementary maps. The main object of this paper will be the study of C ( T ). The idea that a better understanding of the category may give us model theoretic information about T is quite natural: The (semi) group of automorphisms (endomorphisms) of a given structure is often a powerful tool for studying this structure. But certainly, one of the very first questions to be answered is: “to what extent does this category C ( T ) determine T ?” There is some obvious limitation: for example let T 0 be the theory of infinite sets (in a language containing only =) and T 1 the theory, in the language ( =, U ( ν 0 ), f ( ν 0 )) stating that: (1) U is infinite. (2) f is a bijective map from U onto its complement. It is quite easy to see that C ( T 0 ) is equivalent to C ( T 1 ). But, in this case, T 0 and T 1 can be “interpreted” each in the other. To make this notion of interpretation precise, we shall associate with each theory T a category, loosely denoted by T , defined as follows: (1) The objects are the formulas in the given language. (2) The morphisms from into are the formulas such that (i.e. f defines a map from ϕ into ϕ ; two morphisms defining the same map in all models of T should be identified).

Simplicity, and stability in there

2001-06-01

Byunghan Kim

Abstract Firstly, in this paper, we prove that the equivalence of simplicity and the symmetry of forking. Secondly, we attempt to recover definability part of stability theory to simplicity theory. … Abstract Firstly, in this paper, we prove that the equivalence of simplicity and the symmetry of forking. Secondly, we attempt to recover definability part of stability theory to simplicity theory. In particular, using elimination of hyperimaginaries we prove that for any supersimple T . canonical base of an amalgamation class is the union of names of ψ -definitions of , ψ ranging over stationary L -formulas in . Also, we prove that the same is true with stable formulas for an 1-based theory having elimination of hyperimaginaries. For such a theory, the stable forking property holds, too.

Geometric Stability Theory

1996-09-12

Anand Pillay

Abstract This book gives an account of the fundamental results in geometric stability theory, a subject that has grown out of categoricity and classification theory. This approach studies the fine … Abstract This book gives an account of the fundamental results in geometric stability theory, a subject that has grown out of categoricity and classification theory. This approach studies the fine structure of models of stable theories, using the geometry of forking; this often achieves global results relevant to classification theory. Topics range from Zilber-Cherlin classification of infinite locally finite homogenous geometries, to regular types, their geometries, and their role in superstable theories. The structure and existence of definable groups is featured prominently, as is work by Hrushovski. The book is unique in the range and depth of material covered and will be invaluable to anyone interested in modern model theory.

Groupoids, imaginaries and internal covers

2012-04-25

Ehud Hrushovski

Let T be a first-order theory. A correspondence is established between internal covers of models of T and definable groupoids within T. We also consider amalgamations of independent diagrams of … Let T be a first-order theory. A correspondence is established between internal covers of models of T and definable groupoids within T. We also consider amalgamations of independent diagrams of algebraically closed substructures, and find strong relation between covers, uniqueness for 3-amalgamation, existence of 4-amalgamation, imaginaries of T\si, and definable groupoids. As a corollary, we describe the imaginary elements of families of finite-dimensional vector spaces over pseudo-finite fields.

Groupoids, imaginaries and internal covers

2012-01-01

Ehud Hrushovski

Let T be a first-order theory. A correspondence is established between internal covers of models of T and definable groupoids within T. We also consider amalgamations of independent diagrams of … Let T be a first-order theory. A correspondence is established between internal covers of models of T and definable groupoids within T. We also consider amalgamations of independent diagrams of algebraically closed substructures, and find strong relation between covers, uniqueness for 3-amalgamation, existence of 4-amalgamation, imaginaries of T^\si, and definable groupoids. As a corollary, we describe the imaginary elements of families of finite-dimensional vector spaces over pseudo-finite fields.

Chat View PDF

Generalized amalgamation and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" display="inline" overflow="scroll"><mml:mi>n</mml:mi></mml:math>-simplicity

2008-04-23

Byunghan Kim , Alexei Kolesnikov , Akito Tsuboi

CONSTRUCTING THE HYPERDEFINABLE GROUP FROM THE GROUP CONFIGURATION

2006-12-01

Tristram de Piro , Byunghan Kim , Jessica Millar

Chat View PDF

The Lascar groups and the first homology groups in model theory

2017-07-04

Jan Dobrowolski , Byunghan Kim , Junguk Lee

Chat View PDF

Definability in low simple theories

2000-12-01

Ziv Shami

In 1978 Shelah introduced a new class of theories, called simple (see [Shi]) which properly contained the class of stable theories. Shelah generalized part of the theory of forking to … In 1978 Shelah introduced a new class of theories, called simple (see [Shi]) which properly contained the class of stable theories. Shelah generalized part of the theory of forking to the simple context. After approximately 15 years of neglecting the general theory (although there were works by Hrushovski on finite rank with a definability assumption, as well as deep results on specific simple theories by Cherlin, Hrushovski, Chazidakis, Macintyre and Van den dries, see [CH], [CMV], [HP1], [HP2], [ChH]) there was a breakthrough, initiated with the work of Kim ([K1]). Kim proved that almost all the technical machinery of forking developed in the stable context could be generalized to simple case. However, the theory of multiplicity (i.e., the description of the (bounded) set of non forking extensions of a given complete type) no longer holds in the context of simple theories. Indeed, by contrast to simple theories, stable theories share a strong amalgamation property of types, namely if p and q are two “free” complete extensions over a superset of A , and there is no finite equivalence relation over A which separates them, then the conjunction of p and q is consistent (and even free over A .) In [KP] Kim and Pillay proved a weak version of this property for any simple theory, namely “the Independence Theorem for Lascar strong types”. This was a weaker version both because of the requirement that the sets of parameters of the types be mutually independent, as well as the use of Lascar strong types instead of the usual strong types. A very fundamental and interesting problem is whether the independence theorem can be proved for any simple theory, using only the usual strong types. In 1997 Buechler proved ([Bu]) the strong-type version of the independence theorem for an important subclass of simple theories, namely the class of low theories (which includes the class of stable theories and the class of supersimple theories of finite D-rank.)

Notions around tree property 1

2011-03-07

Byunghan Kim , Hyeung-Joon Kim

Totally categorical structures

1989-01-01

Ehud Hrushovski

A first order theory is totally categorical if it has exactly one model in each infinite power. We prove here that every such theory admits a finite language, and is … A first order theory is totally categorical if it has exactly one model in each infinite power. We prove here that every such theory admits a finite language, and is finitely axiomatizable in that language, modulo axioms stating that the structure is infinite. This was conjectured by Vaught. We also show that every <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal alef 0"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi mathvariant="normal">ℵ</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{\aleph _0}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-stable, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal alef 0"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi mathvariant="normal">ℵ</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{\aleph _0}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-categorical structure is a reduct of one that has finitely many models in small uncountable powers. In the case of structures of disintegrated type we nearly find an explicit structure theorem, and show that the remaining obstacle resides in certain nilpotent automorphism groups.

Chat View PDF

Failure ofn-uniqueness: a family of examples

2011-03-07

Elisabetta Pastori , Pablo Spiga

Chat View PDF

Properties of forking in ω-free pseudo-algebraically closed fields

2002-09-01

Zoé Chatzidakis

The study of pseudo-algebraically closed fields (henceforth called PAC) started with the work of J. Ax on finite and pseudo-finite fields [1]. He showed that the infinite models of the … The study of pseudo-algebraically closed fields (henceforth called PAC) started with the work of J. Ax on finite and pseudo-finite fields [1]. He showed that the infinite models of the theory of finite fields are exactly the perfect PAC fields with absolute Galois group isomorphic to , and gave elementary invariants for their first order theory, thereby proving the decidability of the theory of finite fields. Ax's results were then extended to a larger class of PAC fields by M. Jarden and U. Kiehne [21], and Jarden [19]. The final word on theories of PAC fields was given by G. Cherlin, L. van den Dries and A. Macintyre [10], see also results by Ju. Ershov [13], [14]. Let K be a PAC field. Then the elementary theory of K is entirely determined by the following data: • The isomorphism type of the field of absolute numbers of K (the subfield of K of elements algebraic over the prime field). • The degree of imperfection of K . • The first-order theory, in a suitable ω -sorted language, of the inverse system of Galois groups al(L/K) where L runs over all finite Galois extensions of K . They also showed that the theory of PAC fields is undecidable, by showing that any graph can be encoded in the absolute Galois group of some PAC field. It turns out that the absolute Galois group controls much of the behaviour of the PAC fields. I will give below some examples illustrating this phenomenon.

Toward classifying unstable theories

1996-08-01

Saharon Shelah

On ◁∗-maximality

2003-12-22

Mirna Džamonja , Saharon Shelah

The diameter of a Lascar strong type

2003-01-01

Ludomir Newelski

We prove that a type-definable Lascar strong type has finite diameter. We also answer some other questions from [1] on Lascar strong types. We give some applications on subgroups of … We prove that a type-definable Lascar strong type has finite diameter. We also answer some other questions from [1] on Lascar strong types. We give some applications on subgroups of type-definable groups.

Chat View PDF

The Structure of Compact Groups

2013-08-16

Karl H. Hofmann , Sidney A. Morris

The subject matter of compact groups is frequently cited in fields like algebra, topology, functional analysis, and theoretical physics. This book serves the dual purpose of providing a textbook on … The subject matter of compact groups is frequently cited in fields like algebra, topology, functional analysis, and theoretical physics. This book serves the dual purpose of providing a textbook on it for upper level graduate courses or seminars, and of serving as a source book for research specialists who need to apply the structure and representation theory of compact groups. After a gentle introduction to compact groups and their representation theory, the book presents self-contained courses on linear Lie groups, on compact Lie groups, and on locally compact abelian groups. Separate appended chapters contain the material for courses on abelian groups and on category theory. However, the thrust of the book points in the direction of the structure theory of not necessarily finite dimensional, nor necessarily commutative, compact groups, unfettered by weight restrictions or dimensional bounds. In the process it utilizes infinite dimensional Lie algebras and the exponential function of arbitrary compact groups. The first edition of 1998 and the second edition of 2006 were well received by reviewers and have been frequently quoted in the areas of instruction and research. For the present new edition the text has been cleaned of typographical flaws and the content has been conceptually sharpened in some places and polished and improved in others. New material has been added to various sections taking into account the progress of research on compact groups both by the authors and other writers. Motivation was provided, among other things, by questions about the structure of compact groups put to the authors by readers through the years following the earlier editions. Accordingly, the authors wished to clarify some aspects of the book which they felt needed improvement. The list of references has increased as the authors included recent publications pertinent to the content of the book.

Hyperimaginaries and automorphism groups

2001-03-01

Daniel Lascar , Anthony L. Pillay

A hyperimaginary is an equivalence class of a type-definable equivalence relation on tuples of possibly infinite length. The notion was recently introduced in [1], mainly with reference to simple theories. … A hyperimaginary is an equivalence class of a type-definable equivalence relation on tuples of possibly infinite length. The notion was recently introduced in [1], mainly with reference to simple theories. It was pointed out there how hyperimaginaries still remain in a sense within the domain of first order logic. In this paper we are concerned with several issues: on the one hand, various levels of complexity of hyperimaginaries, and when hyperimaginaries can be reduced to simpler hyperimaginaries. On the other hand the issue of what information about hyperimaginaries in a saturated structure M can be obtained from the abstract group Aut(M) . In Section 2 we show that if T is simple and canonical bases of Lascar strong types exist in M eq then hyperimaginaries can be eliminated in favour of sequences of ordinary imaginaries. In Section 3, given a type-definable equivalence relation with a bounded number of classes, we show how the quotient space can be equipped with a certain compact topology. In Section 4 we study a certain group introduced in [5], which we call the Galois group of T , develop a Galois theory and make the connection with the ideas in Section 3. We also give some applications, making use of the structure of compact groups. One of these applications states roughly that bounded hyperimaginaries can be eliminated in favour of sequences of finitary hyperimaginaries. In Sections 3 and 4 there is some overlap with parts of Hrushovski's paper [2].

On model-theoretic tree properties

2016-11-22

Artem Chernikov , Nicholas Ramsey

We study model theoretic tree properties ([Formula: see text]) and their associated cardinal invariants ([Formula: see text], respectively). In particular, we obtain a quantitative refinement of Shelah’s theorem ([Formula: see … We study model theoretic tree properties ([Formula: see text]) and their associated cardinal invariants ([Formula: see text], respectively). In particular, we obtain a quantitative refinement of Shelah’s theorem ([Formula: see text]) for countable theories, show that [Formula: see text] is always witnessed by a formula in a single variable (partially answering a question of Shelah) and that weak [Formula: see text] is equivalent to [Formula: see text] (answering a question of Kim and Kim). Besides, we give a characterization of [Formula: see text] via a version of independent amalgamation of types and apply this criterion to verify that some examples in the literature are indeed [Formula: see text].

Chat View PDF

More on 2-chains with 1-shell boundaries in rosy theories

2017-01-01

SunYoung Kim , Junguk Lee

In [4], B. Kim, and the authors classified 2-chains with 1-shell boundaries into either RN (renamable)-type or NR (non renamable)-type 2-chains up to renamability of support of subsummands of a … In [4], B. Kim, and the authors classified 2-chains with 1-shell boundaries into either RN (renamable)-type or NR (non renamable)-type 2-chains up to renamability of support of subsummands of a 2-chain and introduced the notion of chain-walk, which was motivated from graph theory: a directed walk in a directed graph is a sequence of edges with compatible condition on initial and terminal vertices between sequential edges. We consider a directed graph whose vertices are 1-simplices whose supports contain $0$ and edges are plus/minus of $2$-simplices whose supports contain $0$. A chain-walk is a 2-chain induced from a directed walk in this graph. We reduced any 2-chains with 1-shell boundaries into chain-walks having the same boundaries. In this paper, we reduce any 2-chains of 1-shell boundaries into chain-walks of the same boundary with support of size $3$. Using this reduction, we give a combinatorial criterion determining whether a minimal 2-chain is of RN- or NR-type. For a minimal RN-type 2-chains, we show that it is equivalent to a 2-chain of Lascar type (coming from model theory) if and only if it is equivalent to a planar type 2-chain.

Coordinatisation and canonical bases in simple theories

2000-03-01

Bradd Hart , Byunghan Kim , Anand Pillay

In this paper we discuss several generalization of theorems from stability theory to simple theories. Cherlin and Hrushovski, in [2] develop a substitute for canonical bases in finite rank, ω-categorical … In this paper we discuss several generalization of theorems from stability theory to simple theories. Cherlin and Hrushovski, in [2] develop a substitute for canonical bases in finite rank, ω-categorical supersimple theories. Motivated by methods there, we prove the existence of canonical bases (in a suitable sense) for types in any simple theory. This is done in Section 2. In general these canonical bases will (as far as we know) exist only as “hyperimaginaries”, namely objects of the form a / E where a is a possibly infinite tuple and E a type-definable equivalence relation. (In the supersimple, ω-categorical case, these reduce to ordinary imaginaries.) So in Section 1 we develop the general theory of hyperimaginaries and show how first order model theory (including the theory of forking) generalises to hyperimaginaries. We go on, in Section 3 to show the existence and ubiquity of regular types in supersimple theories, ω-categorical simple structures and modularity is discussed in Section 4. It is also shown here how the general machinery of simplicity simplifies some of the general theory of smoothly approximable (or Lie-coordinatizable) structures from [2]. Throughout this paper we will work in a large, saturated model M of a complete theory T . All types, sets and sequences will have size smaller than the size of M . We will assume that the reader is familiar with the basics of forking in simple theories as laid out in [4] and [6]. For basic stability-theoretic results concerning regular types, orthogonality etc., see [1] or [9].

Homology groups of types in stable theories and the Hurewicz correspondence

2014-12-12

John Goodrick , Byunghan Kim , Alexei Kolesnikov

Chat View PDF

Generic expansion and Skolemization in NSOP 1 theories

2018-05-29

Alex Kruckman , Nicholas Ramsey

Chat View PDF

n-Simple theories

2004-08-24

Alexei Kolesnikov

Lascar strong types in some simple theories

1999-06-01

Steven Buechler

Abstract In this paper a class of simple theories, called the low theories is developed, and the following is proved. T heorem . Let T be a low theory, A … Abstract In this paper a class of simple theories, called the low theories is developed, and the following is proved. T heorem . Let T be a low theory, A a set and a, b elements realizing the same strong type over A. Then, a and b realize the same Lasear strong type over A .

An introduction to forking

1979-09-01

Daniel Lascar , Bruno Poizat

The notion of forking has been introduced by Shelah, and a full treatment of it will appear in his book on stability [S1]. The principal aim of this paper is … The notion of forking has been introduced by Shelah, and a full treatment of it will appear in his book on stability [S1]. The principal aim of this paper is to show that it is an easy and natural notion. Consider some well-known examples of ℵ 0 -stable theories: vector spaces over Q , algebraically closed fields, differentially closed fields of characteristic 0; in each of these cases, we have a natural notion of independence: linear, algebraic and differential independence respectively. Forking gives a generalization of these notions. More precisely, if are subsets of some model and c a point of this model, the fact that the type of c over does not fork over means that there are no more relations of dependence between c and than there already existed between c and . In the case of the vector spaces, this means that c is in the space generated by only if it is already in the space generated by . In the case of differentially closed fields, this means that the minimal differential equations of c with coefficient respectively in and have the same order. Of course, these notions of dependence are essential for the study of the above mentioned structures. Forking is no less important for stable theories. A glance at Shelah's book will convince the reader that this is the case. What we have to do is the following. Assuming T stable and given and p a type on , we want to distinguish among the extensions of p to some of them that we shall call the nonforking extensions of p .

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

2015-01-01

Byunghan Kim , SunYoung Kim , Junguk Lee

Chat View PDF

The number of types in simple theories

1999-06-01

Enrique Casanovas

ℵ0-Categorical, ℵ0-stable structures

1985-03-01

Gregory Cherlin , Louise Harrington , A. H. Lachlan

Amalgamation functors and homology groups in model theory

2011-01-01

John Goodrick , Byunghan Kim , Alexei Kolesnikov

Chat View PDF

Generic structures and simple theories

1998-11-01

Zoé Chatzidakis , Anthony L. Pillay

The Structure of Compact Groups

2006-08-22

Karl H. Hofmann , Sidney A. Morris

Dealing with subject matter of compact groups that is frequently cited in fields like algebra, topology, functional analysis, and theoretical physics, this book has been conceived with the dual purpose … Dealing with subject matter of compact groups that is frequently cited in fields like algebra, topology, functional analysis, and theoretical physics, this book has been conceived with the dual purpose of providing a text book for upper level graduate courses or seminars, and of serving as a source book for research specialists who need to apply the structure and representation theory of compact groups. After a gentle introduction to compact groups and their representation theory, the book presents self-contained courses on linear Lie groups, on compact Lie groups, and on locally compact abelian groups. However, the thrust of book points in the direction of the structure theory of infinite dimensional, not necessarily commutative compact groups, unfettered by weight restrictions or dimensional bounds. In the process it utilizes infinite dimensional Lie algebras and the exponential function of arbitrary compact groups. The first edition of 1998 was well received by reviewers and has been frequently quoted in the areas of instruction and research. For the present new edition the text has been cleaned of typographical flaws and some minor inaccuracies of content; it has been edited and improved in various sections. New material has been added in order to reflect ongoing research. In the process of revising the original edition, the integrity of the original section numbering was carefully respected so that citations of material from the first edition remains perfectly viable to the users of this edition.

Tree indiscernibilities, revisited

2013-12-19

Byunghan Kim , Hyeungjoon Kim , Lynn Scow

Chat View PDF

Supersimple theories

2000-09-20

Steven Buechler , Anand Pillay , Frank Olaf Wagner

We prove elimination of hyperimaginaries in supersimple theories. This means that if an equivalence relation on the set of realisations of a complete type (in a supersimple theory) is defined … We prove elimination of hyperimaginaries in supersimple theories. This means that if an equivalence relation on the set of realisations of a complete type (in a supersimple theory) is defined by a possibly infinite conjunction of first order formulas, then it is the intersection of definable equivalence relations.

Definability and definable groups in simple theories

1998-09-01

Anand Pillay

Abstract We continue the study of simple theories begun in [3] and [5]. We first find the right analogue of definability of types. We then develop the theory of generic … Abstract We continue the study of simple theories begun in [3] and [5]. We first find the right analogue of definability of types. We then develop the theory of generic types and stabilizers for groups definable in simple theories. The general ideology is that the role of formulas (or definability) in stable theories is replaced by partial types (or ∞-definability) in simple theories.

Theories without the tree property of the second kind

2013-10-28

Artem Chernikov

Chat View PDF

Model theory of difference fields

1999-04-08

Zoé Chatzidakis , Ehud Hrushovski

A difference field is a field with a distinguished automorphism<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma"><mml:semantics><mml:mi>σ</mml:mi><mml:annotation encoding="application/x-tex">\sigma</mml:annotation></mml:semantics></mml:math></inline-formula>. This paper studies the model theory of existentially closed difference fields. We introduce a dimension theory … A difference field is a field with a distinguished automorphism<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma"><mml:semantics><mml:mi>σ</mml:mi><mml:annotation encoding="application/x-tex">\sigma</mml:annotation></mml:semantics></mml:math></inline-formula>. This paper studies the model theory of existentially closed difference fields. We introduce a dimension theory on formulas, and in particular on difference equations. We show that an arbitrary formula may be reduced into one-dimensional ones, and analyze the possible internal structures on the one-dimensional formulas when the characteristic is<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0"><mml:semantics><mml:mn>0</mml:mn><mml:annotation encoding="application/x-tex">0</mml:annotation></mml:semantics></mml:math></inline-formula>.

Countable models of stable theories

1983-01-01

Anand Pillay

The notion of a normal theory such a theory <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T comma upper I left-parenthesis normal alef 0 comma upper T right-parenthesis equals 1 or greater-than-or-slanted-equals … The notion of a normal theory such a theory <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T comma upper I left-parenthesis normal alef 0 comma upper T right-parenthesis equals 1 or greater-than-or-slanted-equals normal alef 0"> <mml:semantics> <mml:mrow> <mml:mi>T</mml:mi> <mml:mo>,</mml:mo> <mml:mspace width="thickmathspace" /> <mml:mi>I</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi mathvariant="normal">ℵ</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:mrow> <mml:mo>,</mml:mo> <mml:mi>T</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext> or </mml:mtext> </mml:mrow> <mml:mo>⩾</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi mathvariant="normal">ℵ</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">T,\;I({\aleph _0},T) = 1{\text { or }} \geqslant {\aleph _0}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. theorem that for superstable <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T comma upper I left-parenthesis normal alef 0 comma upper T right-parenthesis equals 1 or greater-than-or-slanted-equals normal alef 0"> <mml:semantics> <mml:mrow> <mml:mi>T</mml:mi> <mml:mo>,</mml:mo> <mml:mspace width="thickmathspace" /> <mml:mi>I</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi mathvariant="normal">ℵ</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:mrow> <mml:mo>,</mml:mo> <mml:mi>T</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext> or </mml:mtext> </mml:mrow> <mml:mo>⩾</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi mathvariant="normal">ℵ</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">T,\;I({\aleph _0},T) = 1{\text { or }} \geqslant {\aleph _0}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> stronger than stability but incomparable is introduced, and it is proved that for We also include a short proof of Lachlan’s (The property of normality is to superstability.)

Chat View PDF

On the Number of Countable Models of A Countable Superstable Theory

1973-01-01

A. H. Lachlan

Type-amalgamation properties and polygroupoids in stable theories

2015-05-26

John Goodrick , Byunghan Kim , Alexei Kolesnikov

Chat View PDF