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

Onoj, V. I.; Ursu, L. A.

Mat. Issled. 113, 72-82 (1990).

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 \(n=3\).

Reviewer: I.Corovei (Cluj-Napoca)

MSC:

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

Keywords:

\(n\)-ary loops; identities