ON RECURSIVELY DIFFERENTIABLE K-QUASIGROUPS

Abstract

Recursive differentiability of linear k-quasigroups (k ≥ 2) is studied in the present work. A k-quasigroup is recursively r-differentiable (r is a natu- ral 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(x1, ..., xk) = x1 · x2 · ... · xk, ∀x1, x2, ..., xk ∈ Q, and (Q, ·) 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 [4]. 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.