PARASTROPHICALLY EQUIVALENT QUASIGROUP EQUATIONS

Aleksandar Krapez and Dejan Zivkovic

Abstract: Fedir M. Sokhats'kyi recently posed four problems concerning parastrophic equivalence between generalized quasigroup functional equations. Sava Krstic in his PhD thesis established a connection between generalized quadratic quasigroup functional equations and connected cubic graphs. We use this connection to solve two of Sokhats'kyi's problems, giving also complete characterization of parastrophic cancellability of quadratic equations and reducing the problem of their classification to the problem of classification of connected cubic graphs. Further, we give formulas for the number of quadratic equations with a given number of variables. Finally, we solve all equations with two variables.

Keywords: quasigroup, quadratic functional equation, cubic graph, parastrophic equivalence, parastrophic cancellability

Classification (MSC2000): 39B52; 20N05, 05C25

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