Generating some large filters of quasiorder lattices

For a poset \((P;\le )\) $(P;\le )$ , the quasiorders (AKA preorders) extending the poset order “ \(\le \) $\le$ ” form a complete lattice F, which is a filter in the lattice of all quasiorders of the set P. We prove that if the poset order “ \(\le \) $\le$ ” is small, then F can be generated by few elements.