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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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