  We investigate the connections between clusters and least common ancestors
(LCAs) in directed acyclic graphs (DAGs). We focus on the class of DAGs having
unique least common ancestors for certain subsets of their minimal elements
since these are of interest, particularly as models of phylogenetic networks.
Here, we use the close connection between the canonical k-ary transit function
and the closure function on a set system to show that pre-k-ary clustering
systems are exactly those that derive from a class of DAGs with unique LCAs.
Moreover, we show that k-ary T-systems and k-weak hierarchies are associated
with DAGs that satisfy stronger conditions on the existence of unique LCAs for
sets of size at most k.
