Ultrafilter extensions do not preserve elementary equivalence

Saveliev, Denis I.; Shelah, Saharon

doi:10.1002/malq.201900045

Cited by 6 publications

(5 citation statements)

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“…Ultrafilter extensions of arbitrary n-ary maps (and, more broadly, of first-order models) have been introduced independently in recent works by Goranko [2] and Saveliev [3,14]. Further studies can be found in [15,16,17,18].…”

Section: Application To the Theory Of Ultrafiltersmentioning

confidence: 99%

A note on the Canonical Ramsey Theorem and Ramsey ultrafilters

Polyakov¹

2022

Preprint

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Section: Application To the Theory Of Ultrafiltersmentioning

confidence: 99%

A note on the Canonical Ramsey Theorem and Ramsey ultrafilters

Polyakov¹

2022

Preprint

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“…Both theorems remain true for isomorphic embeddings and some other modeltheoretic interrelations (see [15,27,28]). On the other hand, it is shown in [32] that Theorem 1.7 does not hold for elementary embeddings, moreover, the ultrafilter extensions of a model and its elementary submodel do not need to be elementarily equivalent.…”

Section: Ultrafilter Extensions Of Modelsmentioning

confidence: 99%

“…Another notation was used in[15], where A * was denoted by U(A), and in[27,28,32], where A was denoted by β A.…”

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confidence: 99%

On ultrafilter extensions of first-order models and ultrafilter interpretations

Poliakov

Saveliev

2021

Arch. Math. Logic

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“…Moreover, both theorems remain true for isomorphic embeddings and some other modeltheoretic interrelations (see [1,3,4]). It was shown in [28] that Theorem 2 does not hold for elementary embeddings, moreover, the ultrafilter extensions of a model and an its elementary submodel can be even non-elementarily equivalent.…”

Section: Introductionmentioning

confidence: 99%

“…Another notation was used in[1] where A˚was denoted by UpAq and in[3,4,28] where Ă A was denoted by β β A.…”

mentioning

confidence: 99%

On Two Concepts of Ultrafilter Extensions of First-Order Models and Their Generalizations

Poliakov

Saveliev

2017

Logic, Language, Information, and Computation

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There exist two known concepts of ultrafilter extensions of first-order models, both in a certain sense canonical. One of them [1] comes from modal logic and universal algebra, and in fact goes back to [2]. Another one [3,4] comes from model theory and algebra of ultrafilters, with ultrafilter extensions of semigroups [5] as its main precursor. By a classical fact of general topology, the space of ultrafilters over a discrete space is its largest compactification. The main result of [3,4], which confirms a canonicity of this extension, generalizes this fact to discrete spaces endowed with an arbitrary first-order structure. An analogous result for the former type of ultrafilter extensions was obtained in [6]. Results of such type are referred to as extension theorems.After a brief introduction to this area, we offer a uniform approach to both types of extensions. It is based on the idea to extend the extension procedure itself. We propose a generalization of the standard concept of first-order models in which functional and relational symbols are interpreted rather by ultrafilters over sets of functions and relations than by functions and relations themselves, and an appropriate semantic for generalized models of this form. We provide two specific operations which turn generalized models into ordinary ones, establish necessary and sufficient conditions under which the latter are the two canonical ultrafilter extensions of some ordinary models, and provide a topological characterization of generalized models. Defining a natural concept of homomorphisms between generalized models, we generalize a restricted version of the extension theorem to generalized models. To formulate the full version, we provide even a wider concept of ultrafilter interpretations together with their semantic based on limits of ultrafilters, and show that the former concept can be identified, in a certain way, with a partial case of the latter; moreover, the new concept absorbs the ordinary concept of models. We provide two more specific operations which turn generalized models in the narrow sense into ones in the wide sense, and establish necessary and sufficient conditions under which generalized models in the wide sense are the images of ones in the narrow sense under these operations, and also are two canonical ultrafilter extensions of some ordinary models. Finally, we define homomorphisms between generalized models in the wide sense, and establish for them three full versions of the extension theorem.The results of first three sections of this paper were partially announced in [7].

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Ultrafilter extensions do not preserve elementary equivalence

Abstract: We show that there exist models M1 and M2 such that M1 elementarily embeds into M2 but their ultrafilter extensions β β(M1) and β β(M2) are not elementarily equivalent.

Cited by 6 publications

References 4 publications

A note on the Canonical Ramsey Theorem and Ramsey ultrafilters

A note on the Canonical Ramsey Theorem and Ramsey ultrafilters

On ultrafilter extensions of first-order models and ultrafilter interpretations

On Two Concepts of Ultrafilter Extensions of First-Order Models and Their Generalizations

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