Directed order
From Encyclopedia of Mathematics
A binary relation 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).
Comments
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.
How to Cite This Entry:
Directed order. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Directed_order&oldid=33631
Directed order. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Directed_order&oldid=33631
This article was adapted from an original article by V.I. Ponomarev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article