Semigroup Forum

, Volume 90, Issue 2, pp 374–385 | Cite as

Left large subsets of free semigroups and groups that are not right large

  • Neil Hindman
  • Lakeshia Legette Jones
  • Monique Agnes Peters
RESEARCH ARTICLE
  • 77 Downloads

Abstract

There are several notions of size for subsets of a semigroup \(S\) that originated in topological dynamics and are of interest because of their combinatorial applications as well as their relationship to the algebraic structure of the Stone–Čech compactification \(\beta S\) of \(S\). Among these notions are thick sets, central sets, piecewise syndetic sets, IP sets, and \(\Delta \) sets. Two related notions, namely \(C\) sets and \(J\) sets, arose in the study of combinatorial applications of the algebra of \(\beta S\). If the semigroup is noncommutative, then all of these notions have both left and right versions. In any semigroup, a left thick set must be a right \(J\) set (and of course a right thick set must be a left \(J\) set). We show here that for any free semigroup or free group on more than one generator, there is a set which satisfies all of the left versions of these notions and none of the right versions except \(J\). We also show that for the free semigroup on countably many generators, there is a left \(J\) set which is not a right \(J\) set.

Keywords

Thick Central Piecewise syndetic \(C\)-set \(J\)-set Stone–Čech compactification Free semigroup Free group 

Notes

Acknowledgments

N. Hindman acknowledges support received from the National Science Foundation (USA) via Grant DMS-1160566. L. L. Jones acknowledges support received from the Simons Foundation via Grant 210296.

References

  1. 1.
    Anthony, P.: The smallest ideals in the two natural products on \(\beta S\). Semigroup Forum 48, 363–367 (1994)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Burns, S.: The existence of disjoint smallest ideals in the two natural products on \(\beta S\). Semigroup Forum 63, 191–201 (2001)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    El-Mabhouh, A., Pym, J., Strauss, D.: On the two natural products in a Stone–Čech compactification. Semigroup Forum 48, 255–257 (1994)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Furstenberg, H.: Recurrence in Ergodic Theory and Combinatorical Number Theory. Princeton University Press, Princeton (1981)CrossRefGoogle Scholar
  5. 5.
    Hindman, N., Johnson, J.: Images of \(C\) sets and related large sets under nonhomogeneous spectra, Integers 12A, Article 2 (2012). http://integers-ejcnt.org/vol12b.html
  6. 6.
    Hindman, N., Strauss, D.: Algebra in the Stone–Čech Compactification: Theory and Applications, 2nd edn. Walter de Gruyter & Co., Berlin (2012)Google Scholar
  7. 7.
    Johnson, J.: A new and simpler noncommutative Central Sets Theorem, manuscriptGoogle Scholar
  8. 8.
    Peters, M.: Characterizing differences between the left and right operations on \(\beta S\). Ph.D. Dissertation, Howard University (2013)Google Scholar
  9. 9.
    Rado, R.: Studien zur Kombinatorik. Math. Zeit. 36, 242–280 (1933)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Neil Hindman
    • 1
  • Lakeshia Legette Jones
    • 2
  • Monique Agnes Peters
    • 3
  1. 1.Department of MathematicsHoward UniversityWashingtonUSA
  2. 2.Department of Mathematics and StatisticsUniversity of Arkansas at Little RockLittle RockUSA
  3. 3.Department of MathematicsHoward UniversityWashingtonUSA

Personalised recommendations