Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-10T17:54:22.690Z Has data issue: false hasContentIssue false
  • English
  • Français

STRUCTURAL CONSIDERATIONS OF RAMSEY ALGEBRAS

Part of: Set theory General logic Extremal combinatorics

Published online by Cambridge University Press:  29 October 2021

ZU YAO TEOH Open the ORCID record for ZU YAO TEOH [Opens in a new window]
ZU YAO TEOH*
Affiliation:
SCHOOL OF MATHEMATICAL SCIENCES UNIVERSITI SAINS MALAYSIA11800 USM, MALAYSIA
*
E-mail:teohzuyao@gmail.com
Get access Rights & Permissions [Opens in a new window]

Abstract

Ramsey algebras are an attempt to investigate Ramsey spaces generated by algebras in a purely combinatorial fashion. Previous studies have focused on the basic properties of Ramsey algebras and a few specific examples. In this article, we study the properties of Ramsey algebras from a structural point of view. For instance, we will see that isomorphic algebras have the same Ramsey algebraic properties, but elementarily equivalent algebras need not be so, as expected. We also answer an open question about Cartesian products of Ramsey algebras.

Keywords

Ramsey algebraHindman’s theoremRamsey spaceEllentuck space

MSC classification

Primary: 03E05: Other combinatorial set theory
Secondary: 05D10: Ramsey theory 03B80: Other applications of logic
Type
Article
Information
The Journal of Symbolic Logic , Volume 87 , Issue 4 , December 2022 , pp. 1677 - 1692
Check for updates
Copyright
© The Author(s), 2021. 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.)

References

Carlson, T. J., Some unifying principles in Ramsey theory . Discrete Mathematics , vol. 68 (1988), no. 2, pp. 117–169.CrossRefGoogle Scholar
Chang, C. C. and Keisler, H. J., Model Theory , Studies in Logic and the Foundations of Mathematics, vol. 73, Elsevier, Amsterdam, 1990.Google Scholar
Ellentuck, E., A new proof that analytic sets are Ramsey , this Journal, vol. 39 (1974), no. 01, pp. 163–165.Google Scholar
Erdos, P. and Rado, R., Combinatorial theorems on classifications of subsets of a given set . Proceedings of the London Mathematical Society , vol. 3 (1952), no. 1, pp. 417–439.Google Scholar
Galvin, F. and Prikry, K., Borel sets and Ramsey’s theorem , this Journal, vol. 38 (1973), no. 02, pp. 193–198.Google Scholar
Hindman, N. and Strauss, D., Algebra in the Stone-Čech Compactification: Theory and Applications , De Gruyter Expositions in Mathematics, vol. 27, Walter de Gruyter, Berlin, 1998.CrossRefGoogle Scholar
Keisler, H. J., The ultraproduct construction . Ultrafilters Across Mathematics , vol. 530 (2010), pp. 163–179.CrossRefGoogle Scholar
Milliken, K. R., Ramsey’s theorem with sums or unions . Journal of Combinatorial Theory, Series A , vol. 18 (1975), no. 3, pp. 276–290.CrossRefGoogle Scholar
Rajah, A., Teh, W. C., and Teoh, Z. Y., On the role of associativity in Ramsey algebras . Proceedings of the Mathematical Sciences , vol. 127 (2017), pp. 769–778.CrossRefGoogle Scholar
Sankappanavar, H. P. and Burris, S., A Course in Universal Algebra , Graduate Texts in Mathematics, vol. 78, Springer, New York, 1981.Google Scholar
Silver, J., Every analytic set is Ramsey , this Journal, vol. 35 (1970), no. 01, pp. 60–64.Google Scholar
Teh, W. C., Ramsey algebras and Ramsey spaces , Ph.D. Thesis, The Ohio State University, 2013.Google Scholar
Teh, W. C., Ramsey algebras and strongly reductible ultrafilters . Bulletin of the Malaysian Mathematical Sciences Society , vol. 37 (2014), no. 4, pp. 931–938.Google Scholar
Teh, W. C., Ramsey algebras . Journal of Mathematical Logic , vol. 16 (2016), no. 2, p. 16.CrossRefGoogle Scholar
Teh, W. C., Ramsey algebras and the existence of idempotent ultrafilters . Archive for Mathematical Logic , vol. 55 (2016), nos. 3–4, pp. 475–491.CrossRefGoogle Scholar
Teh, W. C., Ramsey algebras and formal orderly terms . Notre Dame Journal of Formal Logic , vol. 58 (2017), no. 1, pp. 115–125.CrossRefGoogle Scholar
Teh, W. C. and Teoh, Z. Y., Ramsey orderly algebra as a new approach to Ramsey algebras . East-West Journal of Mathematics , vol. 19 (2017), no. 1, pp. 89–72.Google Scholar
Teoh, Z. Y., Ramsey algebras: A Ramseyan combinatorics for universal algebras , Ph.D. Thesis, Universiti Sains Malaysia, 2018.Google Scholar
Teoh, Z. Y. and Teh, W. C., Heterogeneous Ramsey algebras and classification of Ramsey vector spaces . Bulletin of the Malaysian Mathematical Sciences Society , vol. 41 (2016), no. 2, pp. 1011–1028.Google Scholar
Teoh, Z. Y. and Teh, W. C., A Ramsey algebraic study of matrices, Algebra and Discrete Mathematics , vol. 27 (2019), no. 1, pp. 85–98.Google Scholar
Todorcevic, S., Introduction to Ramsey Spaces , Annals of Mathematics Studies, vol. 174, Princeton University Press, Princeton, 2010.CrossRefGoogle Scholar