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 doubly-irreducible elements are preserved by the operation of pruning.