Solution of Belousov's problem (Q2773035)

A local differentiable \(n\)-quasigroup is given on a differentiable manifold by the equation NEWLINE\[NEWLINEx_{n+1}=F(x_1,x_2,\dots,x_n),NEWLINE\]NEWLINE This quasigroup is called reducible if the function \(F(x_1,\dots,x_n)\) has the following form: NEWLINE\[NEWLINEx_{n+1}=g(h(x_1,\dots,x_k),x_{k+1},\dots,x_n),NEWLINE\]NEWLINE (i.e., its operation is reduced to a \(k\)-ary and an \((n+1-k)\)-ary operation, \(2\leq k\leq n-1\)). In his monograph [\(n\)-ary quasigroups, Izdat. ``Shtiintsa'', Kishinev (1972; Zbl 0282.20061)] \textit{V. D. Belousov} posed the following problem: ``Construct examples of irreducible \(n\)-quasigroups, \(n>3\). Do there exist irreducible \(n\)-quasigroups for any \(n>3\)?'' Prior and following this, \textit{V. D. Belousov} and \textit{M. D. Sandik} [in Sib. Mat. Zh. 7, No. 1, 31-54 (1966; Zbl 0199.05201); English transl. in Sib. Math. J. 7, No. 1, 24-42 (1966)] constructed an example of an irreducible \(3\)-quasigroup of order 4. \textit{B. R. Frenkin} [in Mat. Issled. 7, No. 1(23), 150-162 (1972; Zbl 0247.20080)] proved that for any \(n\geq 3\), there exist irreducible \(n\)-quasigroups of order 4. \textit{V. V. Goldberg} [in Sib. Mat. Zh. 16, No. 1, 29-43 (1975; Zbl 0304.20046); English transl. in Sib. Math. J. 16, No. 1, 23-34 (1975) and Sib. Mat. Zh. 17, No. 1, 44-57 (1976; Zbl 0331.53005); English transl. in Sib. Math. J. 17, No. 1, 34-44 (1976)] proved the existence of infinite local irreducible \(n\)-quasigroups for any \(n\geq 3\).NEWLINENEWLINENEWLINEIt is well-known that the theory of \((n+1)\)-webs is equivalent to the theory of local differentiable \(n\)-quasigroups. Goldberg proved that in general an arbitrary \((n+1)\)-web (or a local differentiable \(n\)-quasigroup) is irreducible. One year later, independently, using algebraic methods, \textit{M. M. Glukhov} [in Mat. Issled. 39, 67-72 (1976; Zbl 0374.20079)] proved the existence of infinite irreducible \(n\)-quasigroups for any \(n\geq 3\). For any \(n\geq 3\), \textit{V. V. Borisenko} [in Mat. Issled. 51, 38-42 (1979; Zbl 0432.20058)] constructed examples of irreducible \(n\)-quasigroups of finite composite order \(t>4\). Note that in all these works, no examples of infinite irreducible \(n\)-quasigroups were given.NEWLINENEWLINENEWLINEIn the paper under review, the authors prove that a local \(n\)-quasigroup defined by the equation NEWLINE\[NEWLINEx_{n+1}=F(x_1,\dots,x_n)=\frac{f_1(x_1)+\cdots+f_n(x_n)}{x_1+\cdots+x_n},NEWLINE\]NEWLINE where \(f_i(x_i)\), \(i,j=1,\dots,n\), are arbitrary functions, is irreducible if and only if any two functions \(f_i(x_i)\) and \(f_j(x_j)\), \(i\neq j\), are not both linear homogeneous, or these functions are linear homogeneous but \(\tfrac{f_i(x_i)}{x_i}\neq\tfrac{f_j(x_j)}{x_j}\). This gives a solution to Belousov's problem mentioned above. The authors present the simplest examples of local irreducible and reducible \(n\)-quasigroups. They are coordinate \(n\)-quasigroups of a series of irreducible and reducible \((n+1)\)-webs. The authors came to these examples from the web theory. However, to make this paper accessible for mathematicians not working in web geometry, the authors formulate the main results and their proofs without using the web geometry terminology.