"Preserve the meets" definition on *Bell & Slomson Models and Ultraproducts*

Question

The books says as following (p.21):

Let $\{A_n : n < \omega\}$ be a countable collection of subsets of a Boolean algebra ech having an infumum. Say for $n<\omega$, $a_n = \inf(A_n)$. An ultrafilter $F$ in $B$ is said to preserve the meets iff for $n < \omega$, $h(a_n) = \inf \{ h(a): a \in A_n \}$ where $h$ is the canonical homomorphism of $B$ onto $B/F \cong 2$ .

My question is where does the collection $A_n$ come from? Is the property of "preserving meets" relative to a given collection $A_n$? Or is it expected to work with any collection $\{ A_n\}$? If the latter case, what is the necessity of talking about colletions $\{A_n\}$? Isn't it equivalent (and simpler) to just say that $F$ preserve meets iff for any given collection $A$, $h(\inf A) = \inf \{ h(a): a \in A)\}$?

Answer 1

If actually says:

Say $$\text{for }n<\omega\qquad a_n=\inf(A_n).\tag{$\Pi$}$$ An ultrafilter $F$ in $B$ is said to preserve the meets $(\Pi)$ iff ... ,

so it is clearly referring to the specific meets labelled $(\Pi)$. Theorem $4.10$ that immediately follows also refers specifically to those meets.