a \sqsubseteq b \Leftrightarrow a \supseteq b .
We will call it the reverse poset of filters (or reverse lattice when it is a lattice).
We can denote the lattice operations on as , , , , . By and we denote the minimal and maximal elements of this poset. So set-theoretically, is the filter generated by the union , is the intersection , and similarly for and , while is the power set and is the empty set .