Interpolative fusions

Kruckman, Alex; Tran, Minh Chieu; Walsberg, Erik

doi:10.1142/s0219061321500100

Cited by 6 publications

(13 citation statements)

References 13 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We recall the following results from [KTW21]. Fact 2.3 can be read as saying that T * ∪ is the model companion of T ∪ "relative" to (T i ) i∈I .…”

Section: Preliminariesmentioning

confidence: 99%

“…An existence result. Much of [KTW21] is devoted to developing general conditions ensuring existence of T * ∪ . For the examples in the present paper, we will only need to use one result, Fact 2.6 below.…”

mentioning

confidence: 99%

“…In [KTW21] we obtain an explicit axiomization for T * ∪ under the assumptions of Fact 2.6. This axiomization is ∀∃ when each T i is model complete.…”

mentioning

confidence: 99%

See 2 more Smart Citations

Interpolative fusions II: Preservation results

Kruckman¹,

Tran²,

Walsberg³

2022

Preprint

Self Cite

View full text Add to dashboard Cite

We study interpolative fusion, a method of combining theories T 1 and T 2 in distinct languages in a "generic" way over a common reduct T∩, to obtain a theory T * ∪ . When each T i is model-complete, T * ∪ is the model companion of the union T 1 ∪ T 2 . Our goal is to prove preservation results, i.e., to find sufficient conditions under which model-theoretic properties of T 1 and T 2 are inherited by T * ∪ . We first prove preservation results for quantifier elimination, modelcompleteness, and related properties. We then apply these tools to show that, under mild hypotheses, including stability of T∩, the property NSOP 1 is preserved. We also show that simplicity is preserved under stronger hypotheses on algebraic closure in T 1 and T 2 . This generalizes many previous results; for example, simplicity of ACFA and the random n-hypergraph are both non-obvious corollaries. We also address preservation of stability, NIP, and ℵ 0 -categoricity, and we describe examples which witness that these results are sharp.

show abstract

“…We recall the following results from [KTW21]. Fact 2.3 can be read as saying that T * ∪ is the model companion of T ∪ "relative" to (T i ) i∈I .…”

Section: Preliminariesmentioning

confidence: 99%

mentioning

confidence: 99%

See 1 more Smart Citation

Interpolative fusions II: Preservation results

Kruckman¹,

Tran²,

Walsberg³

2022

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…Problem 7.14. It should be possible to generalize these results on preservation of -dependence to interpolative fusions of theories as studied in [32]. [31] of Artin-Schreier closedness of dependent fields is to show that if is an algebraically independent tuple -that is, trd(F ( )/F ) = + 1 -then is isomorphic over to the additive group, as algebraic groups.…”

mentioning

confidence: 97%

“…The following is an analog of Fact 7.3 in this more general setting. It is essentially from [46], though we refer to [32] here. Namely, let be a complete theory in a language L with a distinguished L(∅)definable set ( ), and let L ′ ⊇ L be a language such that L ′ \ L = { : ∈ } only contains relational symbols.…”

mentioning

confidence: 99%

Onn-dependent groups and fields II

Chernikov

Hempel

2021

Forum of Mathematics, Sigma

View full text Add to dashboard Cite

We continue the study of n-dependent groups, fields and related structures, largely motivated by the conjecture that every n-dependent field is dependent. We provide evidence toward this conjecture by showing that every infinite n-dependent valued field of positive characteristic is henselian, obtaining a variant of Shelah’s Henselianity Conjecture in this case and generalizing a recent result of Johnson for dependent fields. Additionally, we prove a result on intersections of type-definable connected components over generic sets of parameters in n-dependent groups, generalizing Shelah’s absoluteness of $G^{00}$ in dependent theories and relative absoluteness of $G^{00}$ in $2$ -dependent theories. In an effort to clarify the scope of this conjecture, we provide new examples of strictly $2$ -dependent fields with additional structure, showing that Granger’s examples of non-degenerate bilinear forms over dependent fields are $2$ -dependent. Along the way, we obtain some purely model-theoretic results of independent interest: we show that n-dependence is witnessed by formulas with all but one variable singletons; provide a type-counting criterion for $2$ -dependence and use it to deduce $2$ -dependence for compositions of dependent relations with arbitrary binary functions (the Composition Lemma); and show that an expansion of a geometric theory T by a generic predicate is dependent if and only if it is n-dependent for some n, if and only if the algebraic closure in T is disintegrated. An appendix by Martin Bays provides an explicit isomorphism in the Kaplan-Scanlon-Wagner theorem.

show abstract

Generic expansion and Skolemization in NSOP 1 theories

Kruckman

Ramsey

2018

Annals of Pure and Applied Logic

View full text Add to dashboard Cite

We study expansions of NSOP 1 theories that preserve NSOP 1 . We prove that if T is a model complete NSOP 1 theory eliminating the quantifier ∃ ∞ , then the generic expansion of T by arbitrary constant, function, and relation symbols is still NSOP 1 . We give a detailed analysis of the special case of the theory of the generic L-structure, the model companion of the empty theory in an arbitrary language L. Under the same hypotheses, we show that T may be generically expanded to an NSOP 1 theory with built-in Skolem functions. In order to obtain these results, we establish strengthenings of several properties of Kim-independence in NSOP 1 theories, adding instances of algebraic independence to their conclusions.Date: September 18, 2018.

show abstract

Interpolative fusions

Cited by 6 publications

References 13 publications

Interpolative fusions II: Preservation results

Interpolative fusions II: Preservation results

Onn-dependent groups and fields II

Generic expansion and Skolemization in NSOP 1 theories

Contact Info

Product

Resources

About