Belousov equations on ternary quasigroups (Q1332267)

The study of Belousov equations on binary quasigroups was initiated by \textit{V. D. Belousov} [Sib. Mat. Zh. 7, 31-54 (1966; Zbl 0199.05201)]. \textit{A. Krapež} and \textit{M. A. Taylor} [Aequationes Math. 34, 174-185 (1987; Zbl 0645.39006)] showed that every finite set of Belousov equations is equivalent to a single Belousov equation. 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, in some sense, no ``longer'' than any member of the system or the system is equivalent to a pair of equations each with three variables.