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.
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References
Barnsley, M.F.: Fractals everywhere. Academic Press, San Diego (1988)
Binczak, G., Romanowska, A.B., Smith Poset extensions, J.D.H.: Convex sets, and semilattice presentations. Discrete Math. 307, 1–11 (2007)
Bouc, S.: Burnside rings. In: Hazewinkel, M. (ed.) Handbook of Algebra, vol. 2, pp. 739–804. North-Holland, Amsterdam (2000)
Burnside, W.: Theory of Groups of Finite Order. Cambridge University Press, Cambridge (1911). Reprinted by Dover, New York, NY, 1955
Elhamdadi, M., Nelson, S.: Quandles. American Mathematical Society, Providence (2015)
Hutchinson, J.E.: Fractals and self similarity. Indiana Univ. Math. J 30, 713–747 (1981)
Joyce, D.: A classifying invariant of knots, the knot quandle. J. Pure Appl. Algebra 23, 37–65 (1982)
Pudlák, P., Tuma, J.: Every finite lattice can be embedded in a finite partition lattice. Alg. Univ. 10, 74–95 (1980)
Rota, G.-C.: On the foundations of combinatorial theory i: theory of Möbius functions. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 2, 340–368 (1964)
Smith, J.D.H.: Quasigroup homogeneous spaces and linear representations. J. Alg. 241, 193–203 (2001)
Smith, J.D.H.: A coalgebraic approach to quasigroup permutation representations. Alg. Univ. 48, 427–438 (2002)
Smith, J.D.H.: Permutation representations of loops. J. Alg. 264, 342–357 (2003)
Smith, J.D.H.: The Burnside algebra of a quasigroup. J. Alg. 279, 383–401 (2004)
Smith, J.D.H.: Permutation representations of left quasigroups. Alg. Univ. 55, 387–406 (2006)
Smith, J.D.H.: An Introduction to Quasigroups and Their Representations. Chapman and Hall/CRC, Boca Raton (2007)
Smith, J.D.H., Romanowska, A.B.: Post-Modern Algebra. Wiley, New York (1999)
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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
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DOI: https://doi.org/10.1007/s11083-016-9397-9