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On a particular case of the bisymmetric equation for quasigroups. (English) Zbl 1340.39035

The quasigroup-solutions of the generalized bisymmetry equation \[ A(B(x,y),C(u,v))=D(E(x,u),F(y,v)) \] have been determined in the papers, referred by the authors, by showing that they are isotopic to a same abelian group. In this paper, the special case \(B = C = D\), \(A = E\) is discussed and it is proved that the solutions additionally have some additivity and commutativity properties.

MSC:

39B52 Functional equations for functions with more general domains and/or ranges
20N05 Loops, quasigroups
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References:

[1] Aczél J., Belousov V. D., Hosszú M.: Generalized associativity and bisymmetry on quasigroups. Acta Math. Hungar., 11, 127-136 (1960) · Zbl 0090.24301 · doi:10.1007/BF02020630
[2] Belousov V. D.: Some remarks on the functional equation of generalized distributions. Aequationes Math., 1, 54-65 (1968) · Zbl 0157.46402 · doi:10.1007/BF01817557
[3] Krapez A.: Functional equations of generalized associativity, bisymmetry, transitivity and distributivity. Publ. Inst. Math. N.S., 30(44), 81-87 (1982) · Zbl 0508.39013
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