Geodesic
Invertibility criterion of composition of two multiary quasigroups
Commentationes Mathematicae Universitatis Carolinae, Tome 53 (2012) no. 3, pp. 429-445.

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We study invertibility of operations that are composition of two operations of arbitrary arities. We find the criterion for quasigroups and specifications for $T$-quasigroups. For this purpose we introduce notions of perpendicularity of operations and hypercubes. They differ from the previously introduced notions of orthogonality of operations and hypercubes [Belyavskaya G., Mullen G.L.: Orthogonal hypercubes and $n$-ary operations, Quasigroups Related Systems 13 (2005), no. 1, 73--86]. We establish some relationships between these notions and give illustrative examples.
Classification : 05B15, 20N05
Keywords: quasigroup; composition of operations; orthogonal operations; perpendicular operations; hypercube; perpendicular hypercubes; orthogonality of hypercubes; slice; linear quasigroup; $T$-quasigroup 
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Sokhatsky, Fedir M.; Fryz, Iryna V. Invertibility criterion of composition of two multiary quasigroups. Commentationes Mathematicae Universitatis Carolinae, Tome 53 (2012) no. 3, pp. 429-445. http://geodesic.mathdoc.fr/item/CMUC_2012__53_3_a8/