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On quasiidentities of transitive quasigroups. (English) Zbl 0599.20113
It is well known that a quasigroup (Q,\(\cdot)\) is an isotope of an abelian group iff (Q,\(\cdot)\) satisfies the condition of Thomsen \(a\cdot b=d\cdot e\) and \(a\cdot c=f\cdot e\) implies \(d\cdot c=f\cdot b\) and (Q,\(\cdot)\) is an isotope of a group iff (Q,\(\cdot)\) satisfies the conditions of Reidemeister \(a\cdot b=c\cdot d\), \(a\cdot e=c\cdot f\) and \(x\cdot b=y\cdot d\) implies \(x\cdot e=y\cdot f\). Similar to these conditions are the necessary and sufficient conditions we give for a quasigroup (Q,\(\cdot)\) in order that the quasigroup be quasilinear. Such quasigroups are generalizations of linear ones (and also T-quasigroups) that were studied by J. Ježek and T. Kepka.

20N05 Loops, quasigroups
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