I am barely new to order theory and this motivates if the question is trivial.
I understood the definitions of preorder, partially and totally ordered sets and well ordered sets. In particular there is a nice hierarchy among the last three and I think that for a newcomer these are the most natural enviroments to think of, as one is used to the real line and power sets. Moreover these three capture the "natural" idea of order I may think of, at different levels.
On the other hand directed set are less intuitive to me. Clearly do not fit into the hierarchy since they do not libk with partially ordered sets, but totally ordered sets are directed, even if the converse does not hold. Hence I ask: in which sense directed sets generalize totally ordered sets? Which is the intuition behind them? What aspects of order they capture and what motivates the name directed?