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Power sets of \(n\)-ary quasigroups. (English) Zbl 1129.20044

Well-known results with respect to a quasigroup power set (Latin power set) are generalized to the case of \(n\)-ary quasigroups. In this paper are introduced and studied the \((k)\)-powers of a \((k)\)-invertible \(n\)-ary operation and \((k)\)-power sets of \(n\)-ary quasigroups, \(n\geq 2\), \(1\leq k\leq n\). The pairwise orthogonality of such sets is proved, as well as some distinct possibilities of their construction with the help of binary groups and \(n\)-ary groups.

MSC:

20N15 \(n\)-ary systems \((n\ge 3)\)
20N05 Loops, quasigroups
05B15 Orthogonal arrays, Latin squares, Room squares
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