0. Examples

1. Any chain

2.    Pow$(X)$

3.    Div$(n)$ and more generally $\{1,2,\dots \}$ under divisibility

4.    Ideals(Z)$$ and more generally    Ideals(D)$$ for any principal ideal domain $D$

5. Any sublattice of a distributive lattice

6. The lattice of open subsets of a topological space; the lattice of closed subsets of a topological space.

7. $L \times M$, if $L$ and $M$ are distributive

8.    2$^ n$ and more generally    2$^ X$ for any set $X$

9. $L ^ P$ for any distributive lattice $L$ and partially ordered set $P$

10.    Downsets$(P)$ for any partially ordered set $P$. (See Figure .)

(A subset $D \subseteq P$ is a downset if $x \leq y \in D \Rightarrow x \in D$. Observe that unions and intersections of downsets are downsets.)

$\parbox{3in}{ {\bf 11.} \( \mbox{FDL} (3) \), the \lq\lq free distributive lattice on 3 generators'' }$ $\parbox{2in}{ \includegraphics{text/cdir/fdl3.eps} }$

Figure: Downsets$(P)$, with $P$ as indicated

text/cdir/downsets.eps