Exercise 17, § 6

Let $X$ be a lattice (Set Theory, Chapter III, § 1, no. 13). A non-empty subset $F$ of $X$ is called a prefilter if it satisfies the following conditions: (i) whenever $x \in F$ and $y \geq x$, then $y \in F$; (ii) whenever $x \in F$ and $y \in F$, then $\inf(x, y) \in F$; (iii) $F \neq X$. Thus the prefilters on the set of subsets of a set $Y$, ordered by inclusion, are precisely the filters on $Y$. A prefilter $F$ is said to be prime if the relation $\sup(x, y) \in F$ implies that either $x \in F$ or $y \in F$. A prefilter with respect to the ordering opposite to that of $X$ is called a coprefilter on $X$.

a) The set of upper bounds of a subset $A$ of $X$ which does not consist only of the least element of $X$ is a prefilter. Give examples of prefilters which are not sets of upper bounds.

b) Show that if $F$ is a prime prefilter, then $CF$ is a prime coprefilter.

c) Show that if $X$ has a least element $o$, then every prefilter is contained in a maximal prefilter (use Zorn’s lemma, noting that $o$ belongs to no prefilter).

d) Find all prefilters on a linearly ordered set $X$, and show that they are all prime. Hence give examples of prime prefilters which are not maximal.

e) Let $X$ be an ordered set of 5 elements, denoted by $o, 1, a, b, c$, the order relations being $o < a < 1, o < b < 1, o < c < 1$. Show that $X$ is a lattice and that the prefilters other than $\{1\}$ are maximal but not prime.

f) Let $X$ be a lattice having a least element $o$. Show that if $F \subset X$ is a prefilter and $a$ is an element of $X$ which does not belong to $F$, then there is a prefilter containing $F$ and $a$ if and only if $\inf(a, x) \neq o$ for all $x \in F$; the smallest prefilter containing $F$ and $a$ is then the set of all $y \in X$ such that $y \geq \inf(a, x)$ for at least one element $x$ of $F$.