ABSTRACT
This chapter is devoted to the theory of permutation representations or Q-sets of a finite quasigroup Q. In Section 5.1, general finite sets acted upon by a Q-indexed set of Markov matrices are described as Q-IFS or iterated function systems in the sense of fractal geometry. However, the most satisfactory general description is in terms of coalgebras, which are summarized briefly in Appendix C. The Q-IFS are interpreted as certain coalgebras in Section 5.2. Following the technical Section 5.3 describing irreducible coalgebras, the permutation representations or Q-sets of a finite quasigroup Q are then defined in Section 5.4 as the members of the covariety of coalgebras generated by the homogeneous spaces of Q. Section 5.5 introduces the Burnside algebra of a quasigroup, as a direct generalization of the Burnside algebra of a group. Section 5.6 computes the Burnside algebra for the quasigroup of Figure 1.2. Section 5.7 examines the idempotents of the Burnside algebra. Finally, Section 5.8 presents Burnside’s Lemma for quasigroup permutation actions: its proof specializes to a new proof of Burnside’s Lemma for group permutation representations.
