Skip to main content
Log in

Burnside Orders, Burnside Algebras and Partition Lattices

  • Published:
Order Aims and scope Submit manuscript

Abstract

The permutation representation theory of groups has been extended, through quasigroups, to one-sided left (or right) quasigroups. The current paper establishes a link with the theory of ordered sets, introducing the concept of a Burnside order that generalizes the poset of conjugacy classes of subgroups of a finite group. Use of the Burnside order leads to a simplification in the proof of key properties of the Burnside algebra of a left quasigroup. The Burnside order for a projection left quasigroup structure on a finite set is defined by the lattice of set partitions of that set, and it is shown that the general direct and restricted tensor product operations for permutation representations of the projection left quasigroup structure both coincide with the operation of intersection on partitions. In particular, the mark matrix of the Burnside algebra of a projection left quasigroup, a permutation-theoretic concept, emerges as dual to the zeta function of a partition lattice, an order-theoretic concept.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Barnsley, M.F.: Fractals everywhere. Academic Press, San Diego (1988)

    MATH  Google Scholar 

  2. Binczak, G., Romanowska, A.B., Smith Poset extensions, J.D.H.: Convex sets, and semilattice presentations. Discrete Math. 307, 1–11 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bouc, S.: Burnside rings. In: Hazewinkel, M. (ed.) Handbook of Algebra, vol. 2, pp. 739–804. North-Holland, Amsterdam (2000)

    Chapter  Google Scholar 

  4. Burnside, W.: Theory of Groups of Finite Order. Cambridge University Press, Cambridge (1911). Reprinted by Dover, New York, NY, 1955

    MATH  Google Scholar 

  5. Elhamdadi, M., Nelson, S.: Quandles. American Mathematical Society, Providence (2015)

    Book  MATH  Google Scholar 

  6. Hutchinson, J.E.: Fractals and self similarity. Indiana Univ. Math. J 30, 713–747 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  7. Joyce, D.: A classifying invariant of knots, the knot quandle. J. Pure Appl. Algebra 23, 37–65 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  8. Pudlák, P., Tuma, J.: Every finite lattice can be embedded in a finite partition lattice. Alg. Univ. 10, 74–95 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  9. Rota, G.-C.: On the foundations of combinatorial theory i: theory of Möbius functions. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 2, 340–368 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  10. Smith, J.D.H.: Quasigroup homogeneous spaces and linear representations. J. Alg. 241, 193–203 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  11. Smith, J.D.H.: A coalgebraic approach to quasigroup permutation representations. Alg. Univ. 48, 427–438 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  12. Smith, J.D.H.: Permutation representations of loops. J. Alg. 264, 342–357 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  13. Smith, J.D.H.: The Burnside algebra of a quasigroup. J. Alg. 279, 383–401 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  14. Smith, J.D.H.: Permutation representations of left quasigroups. Alg. Univ. 55, 387–406 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  15. Smith, J.D.H.: An Introduction to Quasigroups and Their Representations. Chapman and Hall/CRC, Boca Raton (2007)

    MATH  Google Scholar 

  16. Smith, J.D.H., Romanowska, A.B.: Post-Modern Algebra. Wiley, New York (1999)

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Jonathan D. H. Smith.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Smith, J.D.H. Burnside Orders, Burnside Algebras and Partition Lattices. Order 34, 253–263 (2017). https://doi.org/10.1007/s11083-016-9397-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11083-016-9397-9

Keywords

Navigation