The Post-Gluskin-Hosszú theorem for finite \(n\)-quasigroups and self-invariant families of permutations (Q2812515)

A set \(G\) with an \(n\)-ary operation \(A:G^n\to G\) is called an \(n\)-ary quasigroup if for all \(1\leq i\leq n\) and all fixed elements \(a_1, \ldots, a_{i-1}, a_{i+1},\ldots, a_n, b\), the equation \(A( a_1, \ldots, a_{i-1}, x, a_{i+1},\ldots, a_n)=b\) has a unique solution. If the operation \(A\) is associative the system \((G, A)\) is called an \(n\)-ary group. For the recent case, the structure of \((G,A)\) is determined by the well-known theorem of Hosszú-Gluskin. In the article under review, the author proved a general version of this theorem for a larger class of \(n\)-ary quasigroups, namely the class of 1-weakly invertible on the right \(n\)-ary quasigroups.