Geodesic
Ternary quasigroups and the modular group
Commentationes Mathematicae Universitatis Carolinae, Tome 49 (2008) no. 2, pp. 309-317.

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For a positive integer $n$, the usual definitions of $n$-quasigroups are rather complicated: either by combinatorial conditions that effectively amount to Latin $n$-cubes, or by $2n$ identities on $n+1$ different $n$-ary operations. In this paper, a more symmetrical approach to the specification of $n$-quasigroups is considered. In particular, ternary quasigroups arise from actions of the modular group.
Classification : 08A68, 20N05, 20N15, 20N20
Keywords: quasigroup; ternary quasigroup; $n$-quasigroup; heterogeneous algebra; hyperidentity; modular group; conjugate; parastrophe; time reversal 
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Smith, Jonathan D. H. Ternary quasigroups and the modular group. Commentationes Mathematicae Universitatis Carolinae, Tome 49 (2008) no. 2, pp. 309-317. http://geodesic.mathdoc.fr/item/CMUC_2008__49_2_a12/