Order filter

Remarks

Clause 2 says that $F$ is an upper set. If $P$ is a lattice, clause 3 can be replaced by: $x, y \in F$ implies $x \wedge y \in F$, using the meet. For $a \in P$, the set $\uparrow a = \set{x \in P : a \le x}$ is a filter, the principal filter of $a$. Example: in the power set of $\N$, the cofinite sets $\set{A \subseteq \N : \N \setminus A \text{ is finite}}$ form a proper filter. It contains $\N$, a superset of a cofinite set is cofinite, and $A \cap B$ is cofinite when $A$ and $B$ are, since $\N \setminus (A \cap B) = (\N \setminus A) \cup (\N \setminus B)$. Non-example: in the power set of $\set{1, 2}$, the set $\set{\set{1}, \set{2}, \set{1, 2}}$ satisfies clauses 1 and 2 but not clause 3. The only common lower bound of $\set{1}$ and $\set{2}$ is $\emptyset$, which is not in the set. A filter on a set $X$ in the sense of set filters is a proper filter of the power set of $X$.