Abstract
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, 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 a first-order structure. An analogous result for the former type of ultrafilter extensions was obtained in [6].
Here 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. We provide two specific operations which turn generalized models into ordinary ones, and establish necessary and sufficient conditions under which the latter are the two canonical ultrafilter extensions of some models.
The second author was supported by Grant 16-01-00615 of the Russian Foundation for Basic Research.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Goranko, V.: Filter and ultrafilter extensions of structures: universal-algebraic aspects (2007, preprint)
Jónsson, B., Tarski, A.: Boolean algebras with operators. Part I: Amer. J. Math. 73(4), 891–939 (1951). Part II: ibid. 74(1), 127–162 (1952)
Saveliev, D.I.: Ultrafilter extensions of models. In: Banerjee, M., Seth, A. (eds.) ICLA 2011. LNCS, vol. 6521, pp. 162–177. Springer, Heidelberg (2011). doi:10.1007/978-3-642-18026-2_14
Saveliev, D.I.: On ultrafilter extensions of models. In: Friedman, S.-D., et al. (eds.) The Infinity Project Proceedings CRM Documents 11, Barcelona, pp. 599–616 (2012)
Hindman, N., Strauss, D.: Algebra in the Stone-Čech compactification, 2nd edn. (2012). de Gruyter, W.: Revised and expanded, Berlin-New York
Saveliev, D.I.: On two concepts of ultrafilter extensions of binary relations (2014, preprint)
Čech, E.: On bicompact spaces. Ann. Math. 38(2), 823–844 (1937)
Stone, M.H.: Applications of the theory of Boolean rings to general topology. Trans. Amer. Math. Soc. 41, 375–481 (1937)
Wallman, H.: Lattices and topological spaces. Ann. Math. 39, 112–126 (1938)
Comfort, W.W., Negrepontis, S.: The Theory of Ultrafilters. Springer, Berlin (1974)
Engelking, R.: General topology. Monogr. Matem. 60, Warszawa (1977)
Kochen, S.: Ultraproducts in the theory of models. Ann. Math. 74(2), 221–261 (1961)
Frayne, T., Morel, A.C., Scott, D.S.: Reduced direct products. Fund. Math. 51, 195–228 (1962). 53, 117 (1963)
Chang, C.C., Keisler, H.J.: Model Theory. North-Holland, Amsterdam-London-New York (1973)
Kanamori, A.: The Higher Infinite: Large Cardinals in Set Theory from Their Beginnings, 2nd edn. Springer, Berlin (2005)
Ellis, R.: Lectures on Topological Dynamics. Benjamin, New York (1969)
Saveliev, D.I.: On idempotents in compact left topological universal algebras. Topol. Proc. 43, 37–46 (2014)
Saveliev, D.I.: On Hindman sets (2008, preprint)
Lemmon, E.J.: Algebraic semantics for modal logic. Part II: J. Symb. Logic 31(2), 191–218 (1966)
Lemmon, E.J., Scott, D.S.: An Introduction to Modal Logic. Blackwell, Oxford (1977)
Goldblatt, R.I., Thomason, S.K.: Axiomatic classes in propositional modal logic. In: Crossley, J.N. (ed.) Algebra and Logic. LNM, vol. 450, pp. 163–173. Springer, Heidelberg (1975). doi:10.1007/BFb0062855
van Benthem, J.F.A.K.: Canonical modal logics and ultrafilter extensions. J. Symb. Logic 44(1), 1–8 (1979)
van Benthem, J.F.A.K.: Notes on modal definability. Notre Dame J. Formal Logic 30(1), 20–35 (1988)
Venema, Y.: Model definability, purely modal. In: Gerbrandy, J., et al. (eds.) JFAK. Essays Dedicated to Johan van Benthem on the Occasion on his 50th Birthday, Amsterdam (1999)
Goldblatt, R.I.: Varieties of complex algebras. Ann. Pure Appl. Logic 44, 173–242 (1989)
Saveliev, D.I.: Ultrafilter extensions of linearly ordered sets. Order 32(1), 29–41 (2015)
Acknowledgement
We are indebted to Professor Robert I. Goldblatt who provided some useful historical information concerning the \({}^*\)-extension of relations.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer-Verlag GmbH Germany
About this paper
Cite this paper
Poliakov, N.L., Saveliev, D.I. (2017). On Two Concepts of Ultrafilter Extensions of First-Order Models and Their Generalizations. In: Kennedy, J., de Queiroz, R. (eds) Logic, Language, Information, and Computation. WoLLIC 2017. Lecture Notes in Computer Science(), vol 10388. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-55386-2_24
Download citation
DOI: https://doi.org/10.1007/978-3-662-55386-2_24
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-662-55385-5
Online ISBN: 978-3-662-55386-2
eBook Packages: Computer ScienceComputer Science (R0)