## $$n$$-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

### Keywords:

$$n$$-ary loops; identities
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