On the order of recursive differentiability of finite binary quasigroups (Q6615591)

Let \((Q,A)\) be an \(n\)-ary groupoid and let \(k\) be a positive integer. The \(n\)-ary operation \(A^{(k)}\) is defined on \(Q\) by \[A^{(0)} = A ,\] \[A^{(1)}(x, y) = A(y, A^{(0)}(x, y))\] and \[A^{(t)}(x, y) = A(A^{t-2}(x, y), A^{(t-1)}(x, y))\] for all \(t\geq 2\) and for all \(x, y\in Q\). \N\NA quasigroup \((Q,A)\) is said to be recursively \(r\)-differentiable if the recursive derivatives \(A^{(0)}, A^{(1)}, \ldots, A^{(r)}\) are all quasigroup operations \((r \geq 0)\). Two binary operations \(A\) and \(B\), defined on a non-empty set \(Q\) are said to be orthogonal if the system of equations \(A(x, y) = a\), \(B(x, y) = b\) has a unique solution in \(Q\) for all \(a, b\in Q\). \N\NA system of binary operations \(\{A_1, A_2, \ldots, A_n\}\), \(n \geq 2\) is said to be orthogonal if any two operations are orthogonal. \N\NIn the present article, the author gives an algebraic proof of the following 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 \(r\) of the binary selectors is orthogonal.