Summary: This research concerns the conditions that ensure the quasimonotonicity of the separable operator \(F(x_ 1,x_ 2,\dots, x_ p)= (F_ 1(x_ 1), F_ 2(x_ 2),\dots, F_ p(x_ p))\), where for \(i= 1,2,\dots, p\), \(C_ i\) is an open convex subset of \(\mathbb{R}^{n_ i}\) and \(F_ i: C_ i\to \mathbb{R}^{n_ i}\) is a continuous nonnull operator. It is shown, in particular, that all \(F_ i\), except perhaps one, are monotone. The conditions are given in terms of the monotonicity indices of the operators \(F_ i\), a concept introduced in this paper.