Pseudo-ordered principle for flows strongly monotone with respect to high-rank cones

For C¹-smooth flows strongly monotone with respect to cones of rank k, we prove the pseudo-ordered principle, which concludes that the omega-limit set of a pseudo-ordered orbit is ordered. This gives a positive answer of the open problem posed by Sanchez in 2009. In the case where the rank $ k = 1 $, the pseudo-ordered principle becomes the classical Hirsch's Monotone Convergence Criterion. In the case where the rank $ k = 2 $, the pseudo-ordered principle implies the generic Poincaré-Bendixson theorem: for generic initial points, the omega-limit set is either a nontrivial periodic orbit or a set consisting of equilibria with their potential connected orbits.