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For a poset $(P;\leq)$, the quasiorders (AKA preorders) extending the posetorder "$\leq$" form a complete lattice $F$, which is a filter in the lattice ofall quasiorders of the set $P$. We prove that if the poset order "$\leq$" issmall, then $F$ can be generated by few elements.

Large filters of quasiorder lattices can be generated by few elements