aequationes mathematicae

, Volume 48, Issue 1, pp 35–54 | Cite as

Belousov equations on ternary quasigroups

  • M. A. Taylor
Research Papers


The study of Belousov equations in binary quasigroups was initiated by V. D. Belousov. Krapež and Taylor showed that every finite set of Belousov equations was equivalent to a single Belousov equation which was in some sense no “longer” than any single member of the set. This led to the concept of an irreducible Belousov equation, that is one which is not equivalent to an equation with fewer variables. Krapež and Taylor determined the structure of the irreducible equations by establishing a correspondence between them and specific polynomials overZ2.

In this paper it is shown that the structure of the ternary equations is richer than the binary counterpart, although the main result is similar to the binary case in as far as a system of ternary Belousov equations is equivalent to a single Belousov equation which is no “longer” than any member of the system or the system is equivalent to a pair of equations each with three variables.

AMS (1991) subject classification

Primary 39B52, 20N10 Secondary 20B99 


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  1. [1]
    Belousov, V. D.,Quasigroups with completely reducible balanced equations (Russian). Mat. Issled.83 (1985), 11–25.MathSciNetGoogle Scholar
  2. [2]
    Krapež, A. andTaylor, M. A.,Belousov equations on quasigroups. Aequationes Math.34 (1987), 174–185.Google Scholar
  3. [3]
    Krapež, A. andTaylor, M. A.,Irreducible Belousov equations on quasigroups. Czechoslovak Math. J.43 (1993), 157–175.Google Scholar

Copyright information

© Birkhäuser Verlag 1994

Authors and Affiliations

  • M. A. Taylor
    • 1
  1. 1.Department of Mathematics and StatisticsAcadia UniversityWolfvilleCanada

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