An example of an ultrafilter

Question

This is Theorem 3.5 (pp. 150) of the book "A course in universal algebra." http://www.math.uwaterloo.ca/~snburris/htdocs/UALG/univ-algebra.pdf

Theorem 3.15. Let $\bf B$ be a Boolean algebra.

(a) (Stone) If $a \in B -\{0\}$, then there is a prime ideal $I$ such that $a \notin I$.

(b) If $a \in B-\{1\}$, then there is an ultrafilter $U$ of $\bf B$ with $a\notin U$.

I am having a hard time finding a non-trivial example of $a$, $U$, and $\bf B$ (resp. $a$, $I$, and $\bf B$) that satisfies (b) (resp. (a)).

I will appreciate if someone gives me an example of $a$, $U$, and $\bf B$ (resp. $a$, $I$, and $\bf B$) that satisfies (b) (resp. (a)).

Answer 1

Let $X$ be any infinite set, let $\mathbf B$ be the Boolean algebra consisting of the finite subsets of $X$ and their complements, let $a$ be any finite subset of $X$, and let $U$ consist of all the complements of finite subsets of $X$.

You can get more examples by changing "infinite" and "finite" to "uncountable" and "countable", or to "of cardinality $\geq\kappa$" and "of cardinality $<\kappa$" for your favorite infinite cardinal number $\kappa$.