Belousov equations on quasigroups (Q1103125)

A balanced equation \(w_ 1=w_ 2\) is one in which each variable appears precisely once on both sides. The set of variables in the term \(u\) is denoted by \(<u>\). A balanced equation is called Belousov if for every subterm \(u_ i\) of \(w_ i\) there exists a subterm \(v_ j\) of \(w_ j\) \((i,j=1,2)\) such that \(<u_ i>=<v_ j>\). \textit{V. D. Belousov} [Math. Sb. (N.S.) 70(112), 55-97 (1966; Zbl 0199.05203)] showed that every Belousov equation is equivalent to a finite set of inseparable Belousov equations. In this paper the authors prove that any finite set of Belousov equations is equivalent to a single inseparable Belousov equation with no isolated variables.