The object of the paper are the fundamental relations \(\beta^*\) and \(\gamma^*\) defined on \(n\)-ary polygroups \(P\) as the smallest equivalence relations such that the quotient, \(P/\beta^*\) and \(P/\gamma^*\), by them are \(n\)-ary groups and commutative \(n\)-ary groups, respectively. The authors are studying in depth these relations where the crucial role is played by the kernels of the canonical projections. The concept of \(m\)-ary hyperproduct to introduce a construction of an \(n\)-ary polygroup is also used.