A binary relation on a set with the following properties: 1) if , , then , for any ; 2) for any , always ; and 3) for any there exists a such that , (the Moore–Smith property).
Many authors require a directed order to be a partial order (i.e. to satisfy the condition that and together imply , as well as 1) and 2) above), and also require the underlying set to be non-empty.
Directed order. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Directed_order&oldid=14517