Ultrafilter Extensions of Models

Saveliev, Denis I.

doi:10.1007/978-3-642-18026-2_14

Cited by 11 publications

(45 citation statements)

References 9 publications

(8 reference statements)

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“…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].…”

supporting

confidence: 69%

“…That (ii) implies (i) is trivial since D is a submodel of A. For the converse implication in the case (α), see [3] or [4], Theorem 4.1. The case (β) is obtained from the case (α) as follows.…”

Section: Proofmentioning

confidence: 94%

“…E.g. they remain true for epimorphisms (since for any compact Hausdorff Y , if h : X Ñ Y is such that h" X is dense in Y , then r h : β βX Ñ Y is surjective), as well as for homotopies and isotopies (in sense of [3], [4]), which can be defined for generalized models in the wide sense in the same way as this was done for homomorphisms and embeddings. Also versions for multi-sorted models (having rather many universes X 1 , .…”

Section: Wider Ultrafilter Interpretationsmentioning

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%

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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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supporting

confidence: 69%

Section: Proofmentioning

confidence: 94%

Section: Wider Ultrafilter Interpretationsmentioning

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%

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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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“…This construction was introduced in [1]. The article [2] is an expanded version of [1]; it contains a list of problems, one of which is solved here.…”

Section: Introductionmentioning

confidence: 99%

Ultrafilter extensions do not preserve elementary equivalence

Saveliev

Shelah

2019

Mathematical Logic Qtrly

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More about divisibility in βN

Šobot

2021

Mathematical Logic Qtrly

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We continue the research of an extension ∣∼ of the divisibility relation to the Stone‐Čech compactification βN. First we prove that ultrafilters we call prime actually possess the algebraic property of primality. Several questions concerning the connection between divisibilities in βN and nonstandard extensions of double-struckN are answered, providing a few more equivalent conditions for divisibility in βN. Results on uncountable chains in (βN,true∣∼) are proved and used in a construction of a well‐ordered chain of maximal cardinality. Probably the most interesting result is the existence of a chain of type (R,<) in (βN,true∣∼). Finally, we consider ultrafilters without divisors in double-struckN and among them find the greatest class.

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Ultrafilter Extensions of Models

Cited by 11 publications

References 9 publications

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

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

Ultrafilter extensions do not preserve elementary equivalence

More about divisibility in βN

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