I'm teaching a class on discrete mathematics this quarter and in the course of putting together the midterm exam came across a construction that converts a transitive relation into a strict partial order that I hadn't seen before. If $R$ is a transitive relation over some set $A$, we can define a new relation $S$ over $A$ as follows:
$$ \matrix{xSy & \mbox{if} & xRy \land \lnot(yRx)}$$
Intuitively, this is the relation you get if you take the graph of the binary relation $R$ and define $xSy$ to mean "there's a path from $x$ to $y$ and no path from $y$ back to $x$."
I haven't come across this sort of construction before, but I can see clear parallels between it and other concepts in directed graphs like graph condensations and modules.
Does this construction have a name?