Pruning a poset with veins

We recall some abstract connectivity concepts, and apply them to special chains in partially ordered sets, called veins, that are defined as order-convex chains that are contained in every maximal chain they meet. Veins enable us to define a new partial order on the same underlying set, called the pruning order. The associated pruned poset is simpler than the initial poset, but irreducible, coirreducible, and doublyirreducible elements are preserved by the operation of pruning.