  In this paper, we study the combinatorics of a certain minimal factorization
of the elements in a finite lattice $L$ called the canonical join
representation. The join $\bigvee A =w$ is the canonical join representation of
$w$ if $A$ is the unique lowest subset of $L$ satisfying $\bigvee A=w$ (where
"lowest" is made precise by comparing order ideals under containment). When
each element in $L$ has a canonical join representation, we define the
canonical join complex to be the abstract simplicial complex of subsets $A$
such that $\bigvee A$ is a canonical join representation. We characterize the
class of finite lattices whose canonical join complex is flag, and show how the
canonical join complex is related to the topology of $L$.
