On recursively differentiable k-quasigroups

Abstract

Recursive differentiability of linear $k$-quasigroups $(k\geq 2)$ is studied in the present work. A $k$-quasigroup is recursively $r$-differentiable (r is a natural number) if its recursive derivatives of order up to $r$ are quasigroup operations. We give necessary and sufficient conditions of recursive $1$-differentiability (respectively, $r$-differentiability) of the $k$-group $(Q,B)$, where $B(x_1,..., x_k)=x_1 \cdot x_2 \cdot ... \cdot x_k , \forall x_1 , x_2 ,..., x_k \in Q,$ and $(Q, \cdot)$ is a finite binary group (respectively, a finite abelian binary group). The second result is a generalization of a known criterion of recursive $r$-differentiability of finite binary abelian groups \cite{IzbashSyrbu}. Also we consider a method of construction of recursively $r$-differentiable finite binary quasigroups of high order $r$. The maximum known values of the parameter $r$ for binary quasigroups of order up to 200 are presented.