A generalization of Schröder quasigroups (Q2707965)

A quasigroup satisfying the identity \(xy\cdot yx=x\) is called a Schröder quasigroup. The author generalizes Schröder quasigroups to the \(n\)-ary case. An \(n\)-ary quasigroup \((Q,A)\) satisfying the identity \(A(A(x_1,\dots,x_n),A(x_2,\dots,x_n,x_1),\dots,A(x_n,x_1,\dots,x_{n-1}))=x_1\) is called an \(n\)-ary Schröder quasigroup. Some properties of ternary Schröder quasigroups and \(n\)-ary Schröder quasigroups are determined. The existence of ternary Schröder quasigroups is examined and it is proved that there are no ternary Schröder quasigroups of order 2, 3, 6 but there exist ternary Schröder quasigroups of order \(4^nk\), where \(n\) is a nonnegative integer and \(k\) is an odd integer not divisibile by 3.