Power sets of $n$-ary quasigroups

In the theory of latin squares and in the binary quasigroup theory the notion of a latin power set (a quasigroup power set) is known. These sets have a good property, and namely, they are orthogonal sets. Such sets were studied and methods of their construction were suggested in different articles (see, for example, [1–5]).
In this article we introduce $(k)$-powers of a $k$-invertible $n$-ary operation (with respect to the $k$-multiplication of $n$-ary operations) and $(k)$-power sets of $n$-ary quasigroups, $n\ge 2$, $1\le k\leq n$, prove pairwise orthogonality of such sets and consider distinct posibilities of their construction with the help of binary groups, in particular, using $n$ – $T$-quasigroups and $n$-ary groups.