Exercise 18, § 6

Let $X$ be a distributive lattice (Set Theory, Chapter III, § 1, Exercise 16) which has a least element $o$.

a) If $a$ is an irreducible element of $X$ (Set Theory, Chapter III, § 4, Exercise 7), show that the set of all upper bounds of $a$ is a prime prefilter. Give an example of a prime prefilter which is not a set of upper bounds of an irreducible element of $X$ [cf. Exercise 17 d)].

b) Show that every maximal prefilter on $X$ is prime [use Exercise 17 f)]. Give examples of prime non-maximal prefilters on a distributive lattice [Exercise 17 d)].

c) Show that every prefilter $F$ is the intersection of the prime prefilters which contain $F$ [if $a \notin F$, show that a maximal element $U$ in the set of prefilters which contain $F$ but not $a$ is prime, by using Exercise 17 f)].

d) Let $\Omega$ be the set of prime prefilters on $X$. To each $x \in X$ let correspond the subset $A_x$ of $\Omega$ consisting of prime prefilters $F$ such that $x \in F$. Show that the mapping $x \to A_x$ of $X$ into $\mathfrak{P}(\Omega)$ is injective and order-preserving, that $A_{\operatorname{int}(x,y)} = A_x \cap A_y$ and that