Abstract:
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.
Keywords and phrases:Binary quasigroup, $k$-invertible $n$-ary operation, $n$-ary quasigroup, latin square, $n$-dimensional hypercube, latin power set, quasigroup power set, pairwise orthogonal set of $n$-ary quasigroups.
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