Definition (Filter). $F\subseteq \mathbb{P}$ is called a filter if

• (a) $\forall p,q\in F,\exists r\in F,r\le p,q$
• (b) $\forall p\in F,\forall q\in \mathbb{P},q\ge p⟹q\in F$

If $F$ only has property (a), we call it a filter base, and then

$$\{p\in \mathbb{P}:\exists q\in F,q\le p\}$$

is the filter generated from $F$.