Consider a Boolean algebra $\mathcal{B}:=(B,\leq,\lor,\land,^c,0,1)$ and $\phi \neq X \subseteq \mathcal{P}(B)$ whose elements are filters on $\mathcal{B}$. Show that:
- $\bigcap_{F\in X}F$ is also a filter on $\mathcal{B}$. However, $\bigcup_{F\in X}F$ may not be a filter.
- If $X$ is totally ordered by the inclusion relation $\subseteq$, then $\bigcup_{F\in X}F$ is a filter on $\mathcal{B}$.
I'm working with the definition: $F \subseteq B$ 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 $y\in F$
To start with, I need to show that $\bigcap_{F\in X}F \neq \phi$ - which I'm unable to do. Can we find an element common to all filters on $\mathcal{B}$, which may help us conclude that the intersection is not empty? I'm thinking in this direction because $X$ might as well be the set of all filters on $\mathcal{B}$.
Next, I want to show: if $x,y \in \bigcap_{F\in X}F$ then $x\land y\in \bigcap_{F\in X}F$. This seems easy, since $x,y \in \bigcap_{F\in X}F$ means that $x$ and $y$ are contained in every filter in $X\subseteq\mathcal{P}(B)$, and so is $x\land y$ (property of filters). Similarly for the last property, i.e. if $x \in \bigcap_{F\in X}F$ then $x$ is in every filter in $X$, and we know that filters are upwards closed - so if $x\leq y$ then $y$ is in every filter in $X$ (and hence in $\bigcap_{F\in X}F$)
Next, I want to show that $\bigcup_{F\in X}F$ (non-empty, of course) may not always be a filter - which calls for a counterexample? I'm unable to think of one. So, when will $\bigcup_{F\in X}F$ not be a filter? From the 2nd part, it seems that this may have something to do with ordering?
For the last part, since X is totally ordered, we could probably start off with $X= \{X_1,X_2,...\}$ (X may not be finite, who knows?), and w.l.o.g assume that $X_1 \subseteq X_2 \subseteq ...\subseteq X_i\subseteq X_{i+1}...$ (that's the total ordering defined by inclusion, yes?). How do I take it from here?
TL;DR I have shared my thoughts and work for every part of the question, and it would be a great help if I could get hints or insights that could help me complete my solution (happy to see other solutions also, though)! Thanks!