A binary relation $\leq$ on a set $A$ with the following properties: 1) if $x\leq y$, $y\leq z$, then $x\leq z$, for any $x,y,z\in A$; 2) for any $x\in A$, always $x\leq x$; and 3) for any $x,y\in A$ there exists a $z\in A$ such that $x\leq z$, $y\leq z$ (the Moore–Smith property).
Many authors require a directed order to be a partial order (i.e. to satisfy the condition that $x\leq y$ and $y\leq x$ together imply $x=y$, as well as 1) and 2) above), and also require the underlying set $A$ to be non-empty.
Directed order. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Directed_order&oldid=33631