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

Kim, SunYoung; Lee, Junguk

doi:10.2969/jmsj/06910093

Cited by 3 publications

(10 citation statements)

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“…In Example 2.21, we have already seen that the abelianity assumption cannot be removed, and we present in this section an example showing that the assumption that c is finite cannot be removed either. Lastly, we find a Lie group structure example answering a question raised in [10], which asks whether there exists an RN-pattern minimal 2-chain not equivalent to a Lascar pattern 2-chain having the same boundary.…”

Section: Now Take Any Othermentioning

confidence: 84%

“…3 Also for a,b |= p, if a ≡ L b, then (a,b) is an endpoint pair of a 1-shell which is the boundary of a 2-chain having a Lascar pattern, a kind of an RN-pattern. In [10,Question 4.5], it was asked whether all minimal 2-chains having an RN-pattern with 1-shell boundary should be equivalent to one having a Lascar pattern. It was also noticed that if there is a counterexample, then its length should be at least 7.…”

Section: Rn But Not a Lascar Pattern 2-chainmentioning

confidence: 99%

“…In this subsection, we assume the reader has some familiarity with the notions described in [9,10], and we work with the trivial independence to define independent functors in a strong type (see Definition 0.6), and corresponding 1-shells and 2chains. We now give descriptions of 2-chains having an RN-or a Lascar pattern in terms of balanced walks, rather than original definitions.…”

Section: Rn But Not a Lascar Pattern 2-chainmentioning

confidence: 99%

“…We now give descriptions of 2-chains having an RN-or a Lascar pattern in terms of balanced walks, rather than original definitions. For more combinatorial descriptions for RN-or Lascar pattern 2-chains, see [10]. which is a chain-walk (cf., the proof of [3,Theorem 4.2]).…”

Section: Rn But Not a Lascar Pattern 2-chainmentioning

confidence: 99%

“…In Section 3, we mainly present three counterexamples: a non-G-compact theory where stp(a) is a Lascar type but stp(acl(a)) is not a Lascar type; an example showing that in the above-mentioned isomorphism result in Section 2, the tuple c being finite is essential; and an example answering a question raised in [10], namely a Lie group structure example where an RN-pattern minimal 2-chain is not equivalent to a Lascar pattern 2-chain having the same boundary.…”

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confidence: 99%

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The Relativized Lascar Groups, Type-Amalgamation, and Algebraicity

Dobrowolski¹,

Kim²,

Kolesnikov³

et al. 2021

J. symb. log.

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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 stp(ac) is abelian, and the underlying theory is G-compact, then the Lascar groups of stp(ac) and of 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 stp(ac) is abelian, nor the assumption of c being finite can be removed.

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Section: Now Take Any Othermentioning

confidence: 84%

Section: Rn But Not a Lascar Pattern 2-chainmentioning

confidence: 99%

Section: Rn But Not a Lascar Pattern 2-chainmentioning

confidence: 99%

Section: Rn But Not a Lascar Pattern 2-chainmentioning

confidence: 99%

mentioning

confidence: 99%

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The Relativized Lascar Groups, Type-Amalgamation, and Algebraicity

Dobrowolski¹,

Kim²,

Kolesnikov³

et al. 2021

J. symb. log.

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The Lascar groups and the first homology groups in model theory

Dobrowolski

Kim

Lee

2017

Annals of Pure and Applied Logic

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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 ⌣ | * 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 [3]. 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 ⌣ | * , and can be written simply as H 1 (p).As consequences, in any T , we show that |H 1 (p)| ≥ 2 ℵ0 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.In this paper we study the first homology group of a strong type in any theory.Originally, in [3] and [4], a homology theory only for rosy theories is developed. Namely, given a strong type p in a rosy theory T , the notion of the nth homology group H n (p) depending on thorn-forking independence relation is introduced. Although the homology groups are defined analogously as in singular homology theory in algebraic topology, the (n + 1)th homology group for n > 0 in the rosy theory context has to do with the nth homology group in algebraic topology. For example as in [3], H 2 (p) in stable theories has to do with the fundamental group in topology. 1 2 JAN DOBROWOLSKI, BYUNGHAN KIM, AND JUNGUK LEE theories, H 1 (p) is detecting somewhat endemic properties of p existing only in model theory context.Indeed, in every known rosy example, H n (p) for n ≥ 2 is a profinite abelian group. In [5], it is proved to be so when T is stable under a canonical condition, and conversely, every profinite abelian group can arise in this form. On the other hand, we show in this paper that the first homology groups appear to have distinct features as follows.Let p = tp(a/B) be a strong type over B = acl eq (B) in any theory T . Fix a ternary invariant independence relation ⌣ | * among small sets satisfying finite character, normality, symmetry, transitivity and extension. (There is at least one such relation, by putting A ⌣ | C D for any sets A, C, D.) Then we can analogously define H * 1 (p) depending on ⌣ | * , (which of course is the same as H 1 (p) when ⌣ | * is thorn-independence in rosy T ). In this note, a canonical epimorphism from the Lascar group over B of T to H * 1 (p) is constructed. Indeed, we also introduce the notion of the relativised Lascar group of a type which is proved to be independent from the choice of the monster model of T , and the homomorphism factors th...

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Lascar groups and the first homology groups of strong types in rosy theories

Lee

2015

Preprint

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For a rosy theory, we give a canonical surjective homomorphism from a Lascar group over A = acl eq (A) to a first homology group of a strong type over A, and we describe its kernel by an invariant equivalence relation. As a consequence, we show that the first homology groups of strong types in rosy theories have the cardinalities of one or at least 2 ℵ0 . We give two examples of rosy theories having non trivial first homology groups of strong types over acl eq (∅). In these examples, these two homology groups are exactly isomorphic to their Lascar group over acl eq (∅).

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

Cited by 3 publications

References 4 publications

The Relativized Lascar Groups, Type-Amalgamation, and Algebraicity

The Relativized Lascar Groups, Type-Amalgamation, and Algebraicity

The Lascar groups and the first homology groups in model theory

Lascar groups and the first homology groups of strong types in rosy theories

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