Abstract: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., {\it Orthogonal hypercubes and $n$-ary operations\/}, Quasigroups Related Systems {\bf 13} (2005), no.~1, 73--86]. We establish some relationships between these notions and give illustrative examples.
Keywords: quasigroup, composition of operations, orthogonal operations, perpendicular operations, hypercube, perpendicular hypercubes, orthogonality of hypercubes, slice, linear quasigroup, $T$-quasigroup
AMS Subject Classification: 20N05 05B15