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.

翻译：我们研究了有向无环图（DAGs）中簇与最近公共祖先（LCAs）之间的联系。我们重点关注其特定最小元素子集具有唯一最近公共祖先的一类DAGs，这类图因作为系统发育网络模型而尤为引人关注。本文利用规范k元传递函数与集合系统上闭包函数之间的紧密关联，证明预k元聚类系统恰好源自具有唯一LCAs的DAGs类。此外，我们证明k元T系统和k弱层次结构对应于满足更强条件的DAGs——这些条件要求大小不超过k的集合具有唯一LCAs的存在性。