Interpolation property and homogeneous structures

Gyenis, Zalán

doi:10.1093/jigpal/jzt051

Cited by 3 publications

(18 citation statements)

References 2 publications

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“…We aim to describe structures in which unions of two not explicitly contradictory non‐trivial partial types are realizable. Now we define the Robinson property similarly to . Even with the modified non‐triviality conditions, the Robinson property and the interpolation property remain equivalent (the proof of still works for our case).…”

Section: Interpolation Property and Typesmentioning

confidence: 92%

“…If

T ⊢ φ \to ψ

for some theory T of

{scriptL}_{1} \cup {scriptL}_{2}

then there is a formula ϑ in the language

{scriptL}_{1} \cap {scriptL}_{2}

such that

T ⊢ φ \to ϑ

and

T ⊢ ϑ \to ψ

. Motivated by these results, a similar interpolation property, inside first‐order structures, has been discussed recently by Gyenis in . Let

scriptL

be a language that consists of only finitely many, at most binary relation symbols.…”

Section: Introductionmentioning

confidence: 90%

“…We call a formula

φ (\overset{x}{true}) \in Fm (A)

non‐trivial if any realization of

φ (\bar{x})

is disjoint from A . We must note that there is a subtle difference between the definition of non‐triviality appearing in and the definition here. The former one says that a non‐trivial formula can be realized disjointly from its parameter set, while the latter one, ours, says that every realization is disjoint.…”

Section: Interpolation Property and Typesmentioning

confidence: 99%

“…Gyenis showed that universal and homogeneous structures of such languages satisfy a certain interpolation property if and only if the age of the structure has a special amalgamation property (the so called prescribed amalgamation property ). This paper is a response to the problem left open at the end of . We give a characterization of those universal and homogeneous structures that satisfy the Robinson property without any further restriction on the finite relational language, although with subtle differences on the definitions which we shall point out later.…”

Section: Introductionmentioning

confidence: 99%

“… In this article we discuss a version of the Robinson property studied recently by Gyenis in , and we present a solution to one of his open problems. We say that a first‐order structure

scriptM

satisfies the Robinson property whenever the union of two non‐trivial partial n ‐types over different finite sets is realizable if and only if they are not explicitly contradictory.…”

mentioning

confidence: 99%

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The Robinson property and amalgamations of higher arities

Nyiri

2016

Mathematical Logic Qtrly

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In this article we discuss a version of the Robinson property studied recently by Gyenis in , and we present a solution to one of his open problems. We say that a first‐order structure scriptM satisfies the Robinson property whenever the union of two non‐trivial partial n‐types over different finite sets is realizable if and only if they are not explicitly contradictory. In his article, Gyenis showed that a universal, homogeneous structure over a language that consists of at most binary relation symbols satisfies the Robinson property if and only if its age has a generalized amalgamation property (the so called prescribed amalgamation property, PAP). We go further and define a series of new kinds of amalgamation properties, APn for any natural number n. Using these properties we can characterize all universal and homogeneous structures that satisfy the Robinson property independently from the arities of relation symbols of the language.

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Section: Interpolation Property and Typesmentioning

confidence: 92%

“…If

T ⊢ φ \to ψ

for some theory T of

{scriptL}_{1} \cup {scriptL}_{2}

then there is a formula ϑ in the language

{scriptL}_{1} \cap {scriptL}_{2}

such that

T ⊢ φ \to ϑ

and

T ⊢ ϑ \to ψ

. Motivated by these results, a similar interpolation property, inside first‐order structures, has been discussed recently by Gyenis in . Let

scriptL

be a language that consists of only finitely many, at most binary relation symbols.…”

Section: Introductionmentioning

confidence: 90%

“…We call a formula

φ (\overset{x}{true}) \in Fm (A)

non‐trivial if any realization of

φ (\bar{x})

Section: Interpolation Property and Typesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“… In this article we discuss a version of the Robinson property studied recently by Gyenis in , and we present a solution to one of his open problems. We say that a first‐order structure

scriptM

satisfies the Robinson property whenever the union of two non‐trivial partial n ‐types over different finite sets is realizable if and only if they are not explicitly contradictory.…”

mentioning

confidence: 99%

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The Robinson property and amalgamations of higher arities

Nyiri

2016

Mathematical Logic Qtrly

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The Modelwise Interpolation Property of Semantic Logics

2023

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In this paper we introduce the modelwise interpolation property of a logic that states that whenever $\models\phi\to\psi$ holds for two formulas $\phi$ and $\psi$, then for every model $\mathfrak{M}$ there is an interpolant formula $\chi$ formulated in the intersectionof the vocabularies of \(\phi$ and \(\psi$, such that $\mathfrak{M}\models \phi\to\chi$ and $\mathfrak{M}\models\chi\to\psi$, that is, the interpolant formula in Craig interpolation may vary from model to model. We discuss examples and show that while the $n$-variable fragment of first order logic and difference logic have no Craig interpolation, they both have the modelwise interpolation property. As an application we connect the modelwise interpolation property with the local Beth definability, and we prove that the modelwise interpolation property of an algebraizable logic can be characterized by a weak form of the superamalgamation property of the class of algebras corresponding to the models of the logic. In particular, the class of finite dimensional cylindric set algebras enjoys this weak form of superamalgamation.

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Algebraic Characterization of the Local Craig Interpolation Property

Gyenis

2018

B Sect Log

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Interpolation property and homogeneous structures

Cited by 3 publications

References 2 publications

The Robinson property and amalgamations of higher arities

The Robinson property and amalgamations of higher arities

The Modelwise Interpolation Property of Semantic Logics

Algebraic Characterization of the Local Craig Interpolation Property

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