Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Chains of idempotents in $ \beta\bold N$


Authors: Neil Hindman and Dona Strauss
Journal: Proc. Amer. Math. Soc. 123 (1995), 3881-3888
MSC: Primary 54D35; Secondary 22A15
DOI: https://doi.org/10.1090/S0002-9939-1995-1301502-4
MathSciNet review: 1301502
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We show that any non-minimal idempotent in the semigroup $ (\beta \mathbb{N}, + )$ lies in a sequence of idempotents each smaller than its predecessor and each maximal among all idempotents smaller than its predecessor.


References [Enhancements On Off] (What's this?)

  • [1] J. Auslander, Minimal flows and their extensions, North-Holland, Amsterdam, 1988. MR 956049 (89m:54050)
  • [2] V. Bergelson, W. Deubner, N. Hindman, and H. Lefmann, Rado's Theorem for commutative rings, J. Combin. Theory Ser. A 66 (1994), 68-92. MR 1273292 (95f:05011)
  • [3] J. Berglund, H. Junghenn, and P. Milnes, Analysis on semigroups, Wiley, New York, 1989. MR 999922 (91b:43001)
  • [4] A. Blass and N. Hindman, Sums of ultrafilters and the Rudin-Keisler and Rudin Frolík orders, General Topology and Applications (R. Shortt, ed.), Lecture Notes in Pure and Appl. Math., vol. 123, Dekker, New York, 1990, pp. 59-70. MR 1057624 (91i:03093)
  • [5] R. Ellis, Lectures in topological dynamics, Benjamin, New York, 1969. MR 0267561 (42:2463)
  • [6] A. El-Mabhouh, J. Pym, and D. Strauss, Subsemigroups of $ \beta \mathbb{N}$, Topology Appl. 60 (1994), 87-100. MR 1301065 (95j:22005)
  • [7] L. Gillman and M. Jerison, Rings of continuous functions, van Nostrand, Princeton, NJ, 1960. MR 0116199 (22:6994)
  • [8] N. Hindman, The semigroup $ \beta \mathbb{N}$ and its applications to number theory, The Analytical and Topological Theory of Semigroups (K. Hofmann, J. Lawson, and J. Pym, eds.), Walter deGruyter, Berlin, 1990, pp. 347-360. MR 1072795 (91m:20092)
  • [9] -, Minimal ideals and cancellation in $ \beta \mathbb{N}$, Semigroup Forum 25 (1982), 291-310. MR 679283 (83m:22007)
  • [10] -, Sums equal to products in $ \beta \mathbb{N}$, Semigroup Forum 21 (1980), 221-255. MR 590754 (81m:54040)
  • [11] -, Ultrafilters and combinatorial number theory, Number Theory (Carbondale 1979) (M. Nathanson, ed.), Lecture Notes in Math., vol. 751, Springer-Verlag, Berlin and New York, 1979, pp. 119-184. MR 564927 (81m:10019)
  • [12] N. Hindman, J. van Mill, and P. Simon, Increasing chains of ideals and orbit closures in $ \beta \mathbb{Z}$, Proc. Amer. Math. Soc. 114 (1992), 1167-1172. MR 1089407 (92g:54027)
  • [13] N. Hindman and D. Strauss, Nearly prime subsemigroups of $ \beta \mathbb{N}$, manuscript.
  • [14] W. Ruppert, Compact semitopological semigroups: an intrinsic theory, Lecture Notes in Math., vol. 1079, Springer-Verlag, Berlin, 1984. MR 762985 (86e:22001)
  • [15] D. Strauss, Ideals and commutativity in $ \beta \mathbb{N}$, Topology Appl. 60 (1994), 281-293. MR 1308974 (95j:22007)
  • [16] -, $ {\mathbb{N}^ \ast }$ does not contain an algebraic and topological copy of $ \beta \mathbb{N}$, J. London Math. Soc. (2) 46 (1992), 463-470. MR 1190430 (93k:22002)
  • [17] -, Semigroup structures on $ \beta \mathbb{N}$, Semigroup Forum 44 (1992), 238-244. MR 1141842 (93b:54025)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 54D35, 22A15

Retrieve articles in all journals with MSC: 54D35, 22A15


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1995-1301502-4
Article copyright: © Copyright 1995 American Mathematical Society

American Mathematical Society