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We investigate the connections between clusters and least common ancestors(LCAs) in directed acyclic graphs (DAGs). We focus on the class of DAGs havingunique least common ancestors for certain subsets of their minimal elementssince these are of interest, particularly as models of phylogenetic networks.Here, we use the close connection between the canonical k-ary transit functionand the closure function on a set system to show that pre-k-ary clusteringsystems 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 associatedwith DAGs that satisfy stronger conditions on the existence of unique LCAs forsets of size at most k.

Unique Least Common Ancestors and Clusters in Directed Acyclic Graphs