A graph-theoretic approach to quasigroup cycle numbers

作者：Kerby Brent^*; Smith Jonathan D H

来源：Journal of Combinatorial Theory - Series A, 2011, 118(8): 2232-2245.

DOI：10.1016/j.jcta.2011.05.003

• Latin square
• Quasigroup
• Triangle number
• Schroeder quasigroup
• Triple tournament
• Totally symmetric
• Moufang loop

Norton and Stein associated a number with each idempotent quasigroup or diagonalized Latin square of given finite order n, showing that it is congruent mod 2 to the triangular number T(n). In this paper, we use a graph-theoretic approach to extend their invariant to an arbitrary finite quasigroup. We call it the cycle number, and identify it as the number of connected components in a certain graph, the cycle graph. The congruence obtained by Norton and Stein extends to the general case, giving a simplified proof (with topology replacing case analysis) of the well-known congruence restriction on the possible orders of general Schroeder quasigroups. Cycle numbers correlate nicely with algebraic properties of quasigroups. Certain well-known classes of quasigroups, such as Schroeder quasigroups and commutative Moufang loops, are shown to maximize the cycle number among all quasigroups belonging to a more general class.

• 出版日期2011-11