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\(n\)-ary loops with invertibility properties with one inversion parameter. (Russian) Zbl 0744.20059

Authors’ summary: There are considered the \(n\)-ary loops \((n>2)\) \(Q(\cdot)\) with the identities \[ (\{Jx_ j\}^{i-1}_{j=1},(x^ n_ 1),\{Jx_ j\}^ n_{j=i+1})=x_ i \] for every \(x_ i\in Q\), \(i=1,2,\dots,n\), where \(J\) is a permutation of \(Q\). It is proved that this system of \(n\)-identities is equivalent to a system of two identities when \(n\) is an odd number and to a system of three identities when \(n\) is even. It is constructed an example of such a loop when \(n=3\).

MSC:

20N15 \(n\)-ary systems \((n\ge 3)\)
20N05 Loops, quasigroups
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