What's the meaning 'filtering' and 'chain'? It's about of partially ordered sets. And can you please give me any example?
Definitions:
A preordered set $(I, \leq)$ is directed if every finite subset $F$ of $I$ has an upper bound.
A subset $J$ of a preordered set $(I, \leq)$ is said to be cofinal if for all $i \in I$ there exists $j \in J$ such that $j \geq i$.
A map $f : H \to I$ between two preordered spaces is said to be filtering if for all $i \in I$ there exists $h \in H$ such that $f(k) \geq i$ whenever $k \in H$ satisfies $k \geq h$.
A subset $C$ of $(X, \leq)$ that is totally ordered for the induced preorder is called a chain.