Transitivity of Kim-independence

Kaplan, Itay; Ramsey, Nicholas

doi:10.1016/j.aim.2021.107573

Cited by 10 publications

(26 citation statements)

References 7 publications

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“…Results in this section will be used in the following sections. We make use of ideas from [11] and [14] where the results of this section over models or in other variations are proved earlier. However proofs are considerably simplified, in particular we do not use the notion of ‘tree-Morley sequences’.…”

Section: More Properties Onmentioning

confidence: 99%

“…Contrary to the case of simple theories, an -Morley sequence over A need not be a total -Morley sequence over A . See [11, §3.1] or Example 5.3. We now aim to show Theorem 2.14 about total -Morley sequences, which is analogous to Proposition 2.2 on -Morley sequences.…”

Section: More Properties Onmentioning

confidence: 99%

“…For a set A , denotes . In the rest of this section, we recall definitions and facts from [5, 6, 10, 11, 12, 16], which will be freely used in the paper.…”

Section: Introductionmentioning

confidence: 99%

“…(The facts below are true when the base sets are models in an NSOP theory, regardless of existence. See [10, 11]).…”

Section: Introductionmentioning

confidence: 99%

“…In [6], the results are extended to ‘over arbitrary sets’ when T has existence. Moreover in [11], transitivity of Kim-independence over models is shown in T , and again in [5], that is extended to ‘over any sets’ in T with existence. Consequently more algebraic/combinatorial examples not having SOP are found.…”

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confidence: 99%

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Weak Canonical Bases in Nsop Theories

Kim¹

2021

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We study the notion of weak canonical bases in an NSOP theory T with existence. Given where in , 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 of collinearity of an indiscernible sequence is type-definable. Secondly, we prove that given a total -Morley sequence I in p, the weak canonical base of is , if the hyperimaginary 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 -Morley sequences in p over B. However, while we investigate some examples, we point out that given two weak canonical bases of total -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 theories .

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Section: More Properties Onmentioning

confidence: 99%

Section: More Properties Onmentioning

confidence: 99%

“…For a set A , denotes . In the rest of this section, we recall definitions and facts from [5, 6, 10, 11, 12, 16], which will be freely used in the paper.…”

Section: Introductionmentioning

confidence: 99%

“…(The facts below are true when the base sets are models in an NSOP theory, regardless of existence. See [10, 11]).…”

Section: Introductionmentioning

confidence: 99%

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confidence: 99%

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Weak Canonical Bases in Nsop Theories

Kim¹

2021

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Generic Stability Independence and Treeless Theories

Kaplan,

Ramsey,

Simon

2024

Forum of Mathematics, Sigma

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We initiate a systematic study of generic stability independence and introduce the class of treeless theories in which this notion of independence is particularly well behaved. We show that the class of treeless theories contains both binary theories and stable theories and give several applications of the theory of independence for treeless theories. As a corollary, we show that every binary NSOP $_{3}$ theory is simple.

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Nsop-Like Independence in Aecats

Mark¹

2022

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The classes stable, simple, and NSOP $_1$ in the stability hierarchy for first-order theories can be characterised by the existence of a certain independence relation. For each of them there is a canonicity theorem: there can be at most one nice independence relation. Independence in stable and simple first-order theories must come from forking and dividing (which then coincide), and for NSOP $_1$ theories it must come from Kim-dividing. We generalise this work to the framework of Abstract Elementary Categories (AECats) with the amalgamation property. These are a certain kind of accessible category generalising the category of (subsets of) models of some theory. We prove canonicity theorems for stable, simple, and NSOP $_1$ -like independence relations. The stable and simple cases have been done before in slightly different setups, but we provide them here as well so that we can recover part of the original stability hierarchy. We also provide abstract definitions for each of these independence relations as what we call isi-dividing, isi-forking, and long Kim-dividing.

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Transitivity of Kim-independence

Cited by 10 publications

References 7 publications

Weak Canonical Bases in Nsop Theories

Weak Canonical Bases in Nsop Theories

Generic Stability Independence and Treeless Theories

Nsop-Like Independence in Aecats

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