On the order of recursive differentiability of finite binary quasigroups

Parascovia Syrbu

Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 3 (2023), pp. 103-106

Résumé

The recursive derivatives of an algebraic operation are defined in [1], where they appear as control mappings of complete recursive codes. It is proved in [1], in particular, that the recursive derivatives of order up to $r$ of a finite binary quasigroup $(Q,\cdot )$ are quasigroup operations if and only if $(Q,\cdot )$ defines a recursive MDS-code of length $r+3$. The author of the present note gives an algebraic proof of an equivalent statement: a finite binary quasigroup $(Q,\cdot )$ is recursively $r$-differentiable $(r\geq 0)$ if and only if the system consisting of its recursive derivatives of order up to $r$ and of the binary selectors, is orthogonal. This involves the fact that the maximum order of recursive differentiability of a finite binary quasigroup of order $q$ does not exceed $q-2$.