The Alexandrov topology can be defined for any partially ordered set. Here, a set is open iff it is upwards closed. However, there are other topologies of interest for varied types of partially ordered sets, so I doubt that it is "standard". -JB
The most common and easy to read graphical representation of partial orders is in my opinion not DAGs but Hasse diagrams. In this type of diagrams the direction of the order is implied by the relative positioning of the elements. If there is an arc from x to y and y is above x on the paper then x<=y.
Ok, I took the oportunity to add some other things. -JB
Thanks! Could you also explain the notion of "upwards closed subset"? --AxelBoldt
Which relation does "is a subobject of" refer to? -OJarnef