Pairwise orthogonality of $n$-ary operations

The notions of hypercube and of the orthogonality of two hypercubes were arised in combinatorial analysis. In [11] a connection between $n$-dimensional hypercubes and algebraic $n$-ary operations was established. In this article we use an algebraic approach to the study of orthogonality of two hypercubes (pairwise orthogonality). We give a criterion of orthogonality of two finite $k$-invertible $n$-ary operations, which is used by the research of orthogonality and parastrophe-orthogonality of two $n$-ary $T$-quasigroups. Some examples are given and connection between admissibility and pairwise orthogonality of $n$-ary operations is established.