Skip to main content
SpringerLink
Log in

On ultrafilter extensions of first-order models and ultrafilter interpretations

  • Published:
Archive for Mathematical Logic Aims and scope Submit manuscript

Abstract

There exist two known types of ultrafilter extensions of first-order models, both in a certain sense canonical. One of them (Goranko in Filter and ultrafilter extensions of structures: universal-algebraic aspects, preprint, 2007) comes from modal logic and universal algebra, and in fact goes back to Jónsson and Tarski (Am J Math 73(4):891–939, 1951; 74(1):127–162, 1952). Another one (Saveliev in Lect Notes Comput Sci 6521:162–177, 2011; Saveliev in: Friedman, Koerwien, Müller (eds) The infinity project proceeding, Barcelona, 2012) comes from model theory and algebra of ultrafilters, with ultrafilter extensions of semigroups (Hindman and Strauss in Algebra in the Stone–Čech Compactification, W. de Gruyter, Berlin, 2012) 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 Saveliev (Lect Notes Comput Sci 6521:162–177, 2011; in: Friedman, Koerwien, Müller (eds) The infinity project proceeding, Barcelona, 2012), 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 Saveliev (in On two types of ultrafilter extensions of binary relations. arXiv:2001.02456). Results of such kind are referred to as extension theorems. After a brief introduction, we offer a uniform approach to both types of extensions based on the idea to extend the extension procedure itself. We propose a generalization of the standard concept of first-order interpretations in which functional and relational symbols are interpreted rather by ultrafilters over sets of functions and relations than by functions and relations themselves, and define ultrafilter models with an appropriate semantics for them. We provide two specific operations which turn ultrafilter models into ordinary models, establish necessary and sufficient conditions under which the latter are the two canonical ultrafilter extensions of some ordinary models, and obtain a topological characterization of ultrafilter models. We generalize a restricted version of the extension theorem to ultrafilter models. To formulate the full version, we propose a wider concept of ultrafilter models with their semantics based on limits of ultrafilters, and show that the former concept can be identified, in a certain way, with a particular case of the latter; moreover, the new concept absorbs the ordinary concept of models. We provide two more specific operations which turn ultrafilter models in the narrow sense into ones in the wide sense, and establish necessary and sufficient conditions under which ultrafilter 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 establish three full versions of the extension theorem for ultrafilter models in the wide sense. The results of the first three sections of this paper were partially announced in Poliakov and Saveliev (in: Kennedy, de Queiroz (eds) On two concepts of ultrafilter extensions of first-order models and their generalizations, Springer, Berlin, 2017).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. Another notation was used in [15], where \({\mathfrak {A}}^*\) was denoted by \({{\mathbf {U}}}({\mathfrak {A}})\), and in [27, 28, 32], where \(\widetilde{\,{\mathfrak {A}}\,}\) was denoted by \(\varvec{\beta }\,{\mathfrak {A}}\).

  2. Compare this with non-standard extensions, also used to prove assertions about the extended model, which are elementary; it is unclear, however, whether this technique produces as many results with no known alternative proofs as the technique based on ultrafilter extensions does. Interestingly, a recent paper [7] combines both techniques to obtain results in number theory.

  3. In [25], it was erroneously stated that the set of right continuous maps forms a compact Hausdorff space w.r.t. the pointwise convergence topology; actually, the intended topology was a restricted pointwise convergence topology, as explained in details below.

  4. Ultrafilter models were introduced in [25] under the name of generalized models. In Sect. 4, we shall introduce a wider concept of ultrafilter models (Definition 4.1); the ultrafilter models defined here will be referred to as those “in the narrow sense”.

References

  1. Barwise, J., Feferman, S. (eds.): Model-Theoretic Logics. Perspectives in Mathematical Logic, vol. 8. Springer, Berlin (1985)

    Google Scholar 

  2. Berglund, J., Junghenn, H., Milnes, P.: Analysis on Semigroups. Wiley, New York (1989)

    MATH  Google Scholar 

  3. Blackburn, P., de Rijke, M., Venema, Y.: Modal Logic. Cambridge University Press, Cambridge (2004)

    MATH  Google Scholar 

  4. Čech, E.: On bicompact spaces. Ann. Math. 38(2), 823–844 (1937)

    Article  MathSciNet  Google Scholar 

  5. Chang, C.C., Keisler, H.J.: Model Theory. North-Holland, Amsterdam (1973)

    MATH  Google Scholar 

  6. Comfort, W.W., Negrepontis, S.: The Theory of Ultrafilters. Springer, Berlin (1974)

    Book  Google Scholar 

  7. Di Nasso, M., Luperi Baglini, L.: Ramsey properties of nonlinear Diophantine equations. Adv. Math. 324, 84–117 (2018)

    Article  MathSciNet  Google Scholar 

  8. Ellis, R.: Distal transformation groups. Pac. J. Math. 8, 401–405 (1958)

    Article  MathSciNet  Google Scholar 

  9. Ellis, R.: Lectures on Topological Dynamics. Benjamin, New York (1969)

    MATH  Google Scholar 

  10. Engelking, R.: General topology. Monogr. Matem. vol. 60, Warszawa (1977)

  11. Frayne, T., Morel, A.C., Scott, D.S.: Reduced direct products. Fund. Math. 51(3), 195–228 (1962)

    Article  MathSciNet  Google Scholar 

  12. Frayne, T., Morel, A.C., Scott, D.S.: Correction to the paper “Reduced direct products”. Fund. Math. 53(1), 117 (1963)

    MathSciNet  Google Scholar 

  13. Goldblatt, R.I., Thomason, S.K.: Axiomatic classes in propositional modal logic. In: Crossley, J. N. (ed.). Algebra and Logic. Lecture Notes in Mathematics, vol. 450, pp. 163–173 (1975)

  14. Goldblatt, R.I.: Varieties of complex algebras. Ann. Pure Appl. Logic 44, 173–242 (1989)

    Article  MathSciNet  Google Scholar 

  15. Goranko, V.: Filter and ultrafilter extensions of structures: universal-algebraic aspects. Preprint (2007)

  16. Hindman, N., Strauss, D.: Algebra in the Stone–Čech Compactification, 2nd edn., revised and expanded, W. de Gruyter, Berlin (2012)

  17. Hindman, N., Strauss, D.: Topological properties of some algebraically defined subsets of \({\varvec {\beta }}{\mathbb{N}}\). Topology and its Application 220, 43–49 (2017)

    Article  MathSciNet  Google Scholar 

  18. Hindman, N., Strauss, D.: Sets and mapping in \({\varvec {\beta }}S\) which are not Borel. N. Y. J. Math. 24, 689–701 (2018)

    MathSciNet  MATH  Google Scholar 

  19. Jónsson, B., Tarski, A.: Boolean algebras with operators. Part I. Am. J. Math. 73(4), 891–939 (1951)

    Article  Google Scholar 

  20. Jónsson, B., Tarski, A.: Boolean algebras with operators. Part II. Am. J. Math. 74(1), 127–162 (1952)

    Article  Google Scholar 

  21. Kanamori, A.: The Higher Infinite: Large Cardinals in Set Theory from Their Beginnings, 2nd edn. Springer, Berlin (2005)

    MATH  Google Scholar 

  22. Kochen, S.: Ultraproducts in the theory of models. Ann. Math. 74(2), 221–261 (1961)

    Article  MathSciNet  Google Scholar 

  23. Lemmon, E.J.: Algebraic semantics for modal logic. Part II. J. Symb. Logic 31(2), 191–218 (1966)

    Article  Google Scholar 

  24. Lemmon, E.J., Scott, D.S.: An Introduction to Modal Logic. Blackwell, Oxford (1977)

    Google Scholar 

  25. Poliakov, N.L., Saveliev, D.I.: LLIC, Lecture Notes in Computer Science. In: Kennedy, J., de Queiroz, R.J.G.B. (eds.) On Two Concepts of Ultrafilter Extensions of First-order Models and Their Generalizations, vol. 10388, pp. 336–348. Springer, Berlin (2017)

    MATH  Google Scholar 

  26. Saveliev, D.I.: On Hindman sets. Preprint (2008)

  27. Saveliev, D.I.: Ultrafilter extensions of models. Lect. Notes Comput. Sci. 6521, 162–177 (2011)

    Article  MathSciNet  Google Scholar 

  28. Saveliev, D.I.: On ultrafilter extensions of models. The Infinity Project Proceeding. In: S.-D. Friedman, M. Koerwien, M. M. Müller (eds.), CRM Documents 11, Barcelona, pp. 599–616 (2012)

  29. Saveliev, D.I.: On idempotents in compact left topological universal algebras. Topol. Proc. 43, 37–46 (2014)

    MathSciNet  MATH  Google Scholar 

  30. Saveliev, D.I.: Ultrafilter extensions of linearly ordered sets. Order 32(1), 29–41 (2015)

    Article  MathSciNet  Google Scholar 

  31. Saveliev, D.I.: On two types of ultrafilter extensions of binary relations. arXiv:2001.02456

  32. Saveliev, D.I., Shelah, S.: Ultrafilter extensions do not preserve elementary equivalence. Math. Log. Q. 65(4), 511–516 (2019)

    Article  MathSciNet  Google Scholar 

  33. Shelah, S.: There are just four second-order quantifiers. Israel J. Math. 15, 282–300 (1973)

    Article  MathSciNet  Google Scholar 

  34. Stone, M.H.: Applications of the theory of Boolean rings to general topology. Trans. Am. Math. Soc. 41, 375–481 (1937)

    Article  MathSciNet  Google Scholar 

  35. van Benthem, J.F.A.K.: Notes on modal definability. Notre Dame J. Formal Logic 30(1), 20–35 (1988)

    MathSciNet  MATH  Google Scholar 

  36. van Benthem, J.F.A.K.: Canonical modal logics and ultrafilter extensions. J. Symb. Logic 44(1), 1–8 (1979)

    Article  MathSciNet  Google Scholar 

  37. Venema, Y.: Model definability, purely modal. In: J. Gerbrandy et al. (eds). JFAK. Essays Dedicated to Johan van Benthem on the Occasion on his 50th Birthday. Amsterdam (1999)

  38. Wallman, H.: Lattices and topological spaces. Ann. Math. 39, 112–126 (1938)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

We would like to express our gratitude to Professors Robert I. Goldblatt and Neil Hindman who provided us some useful historical information. We are also indebted to two anonymous referees for some critical remarks and suggestions.

Author information

Authors and Affiliations

  1. HSE University, Pokrovsky Boulevard 11, T423, Moscow, Russia, 109028

    Nikolai L. Poliakov

  2. Institute for Information Transmission Problems, Russian Academy of Sciences, Bolshoy Karetny Lane 19 Building 1, Moscow, Russia, 127051

    Denis I. Saveliev

Authors
  1. Nikolai L. Poliakov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Denis I. Saveliev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Denis I. Saveliev.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Poliakov, N.L., Saveliev, D.I. On ultrafilter extensions of first-order models and ultrafilter interpretations. Arch. Math. Logic 60, 625–681 (2021). https://doi.org/10.1007/s00153-021-00783-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00153-021-00783-6

Keywords

Mathematics Subject Classification

Search

Navigation