Consider a Boolean algebra $\mathcal{B}:=(B,\leq,\lor,\land,^c,0,1)$. Let $\mathcal{F}_\mathcal{B}$ denote the set of all filters on $\mathcal{B}$, and let $\mathcal{F'}_\mathcal{B}$ denote the set of all principal filters on $\mathcal{B}$. Is $\mathcal{F}_\mathcal{B}$ = $\mathcal{F'}_\mathcal{B}$?
$F \subseteq B$ is a filter if -
- $F \neq \phi$
- If $x,y \in F$ then $x\land y\in F$
- If $x\in F$ and $x\leq y$ then $y\in F$
Moreover, $F(a) = \{x\in B: a\leq x\}$ denotes the principal filter w.r.t $a\in B$, i.e. the smallest filter containing $a$.
When is $\mathcal{F}_\mathcal{B}$ = $\mathcal{F'}_\mathcal{B}$? Does it have something to do with whether or not the $\mathcal{B}$ is complete? I don't know where to start (except I'm trying to look for counterexamples), and I'd appreciate any help. Thanks!