## Many-place separable quasigroups with invertibility properties.(Russian)Zbl 0738.20062

First, some definitions. An $$n$$-ary quasigroup $$(Q,f)$$, $$f: Q\times Q\times\dots\times Q\to Q$$, is said to be separable [Reviewer’s comment: the term “reducible” is more applied.] if $$f$$ has a nontrivial composition, $$f=g_ 1\circ g_ 2$$. An $$n$$-ary quasigroup $$(Q,f)$$ is said to be a parastroph of $$Q$$ if $$x_{n+1}=f(x_ 1,\dots,x_ n)\Leftrightarrow\sigma(x_{n+1})=\sigma f(x_{\sigma(1)},\dots,x_{\sigma(n)})$$, where $$\sigma$$ is a permutation. The parastroph $$f$$ is called principal, if $$\sigma(x_{n+1})=x_{n+1}$$. Finally, we say that the quasigroup $$(Q,\bar f)$$ is isotopic to $$(Q,f)$$ if there exist $$n+1$$ permutations $$\alpha_ i: Q\to Q$$ such that $$\bar f(\alpha_ 1(x_ 1),\dots,\alpha_ n(x_ n))=\alpha_{n+1}\circ f(x_ 1,\dots,x_ n)$$. It is proved that the quasigroup $$(Q,f)$$ is separable and is isotopic to each of its principal parastrophs iff an Abelian group $$(Q,+)$$ and $$n$$ permutations $$\alpha_ i: Q\to Q$$ exist such that $$f(x_ 1,\dots,x_ n)=\alpha_ 1(x_ 1)+\dots+\alpha_ n(x_ n)$$. The result for some classes of $$n$$-quasigroups is given in detail.

### MSC:

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