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On the classification of functional equations on quasigroups

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Abstract

We consider functional equations over quasigroup operations. We prove that every quadratic parastrophically uncancelable functional equation for four object variables is parastrophically equivalent to the functional equation of mediality or the functional equation of pseudomediality. The set of all solutions of the general functional equation of pseudomediality is found and a criterion for the uncancelability of a quadratic functional equation for four object variables is established.

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REFERENCES

  1. V. D. Belousov, n-ary Quasigroups [in Russian], Shtiintsa, Kishinev (1972).

    Google Scholar 

  2. M. A. Taylor, “A generalization of a theorem of Belousov,” Bull. London Math. Soc., 10, No.3, 285–286 (1978).

    Google Scholar 

  3. A. Krapež, “Strictly quadratic functional equations on quasigroups. I,” Publ. Inst. Math., No. 29(43), 1–17 (1979).

    Google Scholar 

  4. V. D. Belousov, “Quasigroups with completely cancelable balanced identities,” in: Studies on the Theory of Binary and n-ary Quasigroups [in Russian], Shtiintsa, Kishinev (1985), pp. 11–25.

    Google Scholar 

  5. J. Duplák, “Identities and deleting maps on quasigroups,” Math. Inst. Czech. Acad. Sci., 38, No.1, 1–7 (1988).

    Google Scholar 

  6. F. M. Sokhats’kyi, “On isotopes of groups. I,” Ukr. Mat. Zh., 47, No.10, 1387–1398 (1995).

    Article  Google Scholar 

  7. J. Duplák, “A parastrophic equivalence in quasigroups,” Quasigroups Rel. Syst., 7, 7–14 (2000).

    MathSciNet  Google Scholar 

  8. V. D. Belousov, “Balanced identities in quasigroups,” Mat. Sb., 70, No.1, 55–97 (1966).

    Google Scholar 

  9. V. D. Belousov, “One theorem on balanced identities,” Mat. Issled., 71, 22–24 (1983).

    Google Scholar 

  10. V. D. Belousov, “Equation of general mediality,” Mat. Issled., 39, 21–31 (1976).

    Google Scholar 

  11. F. M. Sokhats’kyi and R. F. Koval’, “A classification of functional equations for four variables,” in: Abstracts of the 3rd International Algebraic Conference in Ukraine [in Ukrainian], Sumy (2001), pp. 254–255.

  12. F. M. Sokhats’kyi and R. F. Koval’, “A classification of general four-variable quadratic parastrophically uncancelable functional equations on quasigroups,” in: Abstracts of the 4th International Algebraic Conference in Ukraine (Lviv, August 4–9, 2003) [in Ukrainian], Lviv (2003).

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Authors and Affiliations

  1. Vinnitsa Pedagogic University, Vinnitsa

    F. M. Sokhats’kyi

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  1. F. M. Sokhats’kyi
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 56, No. 9, pp. 1259–1266, September, 2004.

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Sokhats’kyi, F.M. On the classification of functional equations on quasigroups. Ukr Math J 56, 1499–1508 (2004). https://doi.org/10.1007/s11253-005-0129-3

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  • DOI: https://doi.org/10.1007/s11253-005-0129-3

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