Hostname: page-component-77f85d65b8-6c7dr Total loading time: 0 Render date: 2026-04-10T11:44:35.275Z Has data issue: false hasContentIssue false
  • English
  • Français

CONTINUOUS SENTENCES PRESERVED UNDER REDUCED PRODUCTS

Part of: Model theory

Published online by Cambridge University Press:  07 September 2020

ISAAC GOLDBRING  and
H. JEROME KEISLER
ISAAC GOLDBRING
Affiliation:
DEPARTMENT OF MATHEMATICS, STATISTICS, AND COMPUTER SCIENCE DEPARTMENT OF MATHEMATICS UNIVERSITY OF CALIFORNIA, IRVINE 340 ROWLAND HALL (BLDG.# 400), IRVINE, CA 92697-3875, USA E-mail: isaac@math.uci.edu
H. JEROME KEISLER
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF WISCONSIN 480 LINCOLN DRIVE, MADISON WI 53706, USA E-mail: keisler@math.wisc.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

Answering a question of Cifú Lopes, we give a syntactic characterization of those continuous sentences that are preserved under reduced products of metric structures. In fact, we settle this question in the wider context of general structures as introduced by the second author.

Keywords

reduced productscontinuous logic

MSC classification

Primary: 03C20: Ultraproducts and related constructions

Information

Type
Article
Information
The Journal of Symbolic Logic , Volume 87 , Issue 2 , June 2022 , pp. 649 - 681
Check for updates
Copyright
© The Author(s), 2020. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

CaicedoX. Caicedo, X., Maximality of continuous logic, Beyond First Order Model Theory (Iovino, J., editor), CRC Press, Boca Raton, FL, 2017, 105–130.CrossRefGoogle Scholar
Chang, C. C. and Keisler, H. J., Continuous Model Theory , Annals of Mathematics Studies, Princeton, NJ, 1966.10.1515/9781400882052CrossRefGoogle Scholar
Chang, C. C. and Keisler, H. J., Model Theory , Dover Publications, Mineola, NY, 2012.Google Scholar
Farah, I. and Shelah, S., Rigidity of continuous quotients . Journal of the Institute of Mathematics of Jussieu , vol. 15 (2016), 1–28.CrossRefGoogle Scholar
Galvin, F., Horn sentences . Annals of Mathematical Logic , vol. 1 (1970), 389–422.CrossRefGoogle Scholar
Ghasemi, S., Reduced products of metric structures: A metric Feferman–Vaught theorem . this Journal , vol. 81 (2016), 856–875.Google Scholar
Goldbring, I. and Hart, B., On the theories of McDuffs II ${}_1$ factors . International Mathematics Research Notices , vol. 27 (2017), 5609–5628.Google Scholar
Henson, C. Ward. Nonstandard hulls in Banach spaces, Israel Journal of Mathematics , vol. 25 (1976), 108–144.CrossRefGoogle Scholar
Henson, C. Ward and Iovino, J., Ultraproducts in analysis . Analysis and Logic , London Mathematical Society Lecture Note Series, vol. 262 (2002), 1–113.Google Scholar
Keisler, H. J., Ultraproducts and elementary classes. Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen. Series A (1961), 477–495.CrossRefGoogle Scholar
Keisler, H. J., Reduced products and Horn classes . Transactions of the American Mathematical Society , vol. 117 (1965), 307–328.CrossRefGoogle Scholar
Keisler, H. J., Model theory for real–valued structures. Beyond Second Order Model Theory , to appear. Ed. by Iovino, Jose.Google Scholar
Kunen, K., Ultrafilters and independent sets . Transactions of the American Mathematical Society , vol. 172 (1972), 199–206.CrossRefGoogle Scholar
Lopes, V. C., Reduced products and sheaves of metric structures . Mathematical Logic Quarterly , vol. 59 (2013), 219–229.CrossRefGoogle Scholar
Shelah, S., Every two elementarily equivalent models have isomorphic ultrapowers . Israel Journal of Mathematics , vol. 10 (1972), 224–233.CrossRefGoogle Scholar
Yaacov, I. B., Positive model mheory and compact abstract theories . Journal of Mathematical Logic , vol. 3 (2003), 85–118.10.1142/S0219061303000212CrossRefGoogle Scholar
Yaacov, I. B., On theories of random variables . Israel Journal of Mathematics , vol. 194 (2013), 957–1012.10.1007/s11856-012-0155-4CrossRefGoogle Scholar
Yaacov, I. B., Berenstein, A., Henson, C. W., and Usvyatsov, A., Model theory for metric structures, Model Theory with Applications to Algebra and Analysis , (Chatzidakis, Z., Macpherson, D., Pillay, A., and Wilkie, A., editors), London Mathematical Society Lecture Note Series, vol. 350, Cambridge, Cambridge University Press, 2008, 315–427.10.1017/CBO9780511735219.011CrossRefGoogle Scholar