A generalization of Schröder quasigroups

Abstract

A generalization of Schröder quasigroups (quasigroups satisfying the identity xy · yx = x) to the n-ary case is considered. An n-ary quasigroup (Q, A) satisfying the identity A(A(x 1 , . . . , x n ), A(x 2 , . . . , x n , x 1 ) , . . . , A(x n , x 1 , . . . , x n-1 )) = x 1 is called an n-ary Schröder quasigroup (nSQ). Some properties of ternary SQs (TSQs) and nSQs are determined. Every nSQ of order v is self-orthogonal and also it defines an orthogonal set of n (n - 1)-ary quasigroups of order v. The existence of TSQs is examined and it is proved that there are no TSQs of order 2,3,6, but there exist TSQs of order v = 4 α k, where α is a nonnegative integer and k is an odd integer not divisible by 3. Every TSQ of order n defines an n 3 × 6 orthogonal array (OA). Conjugations leaving invariant an OA associated with an TSQ are also investigated.