Directed set
NONEMPTY SET; GENERALIZATION OF NONEMPTY TOTALLY ORDERED SETS
Upward directed set; Upward-directed set; Directed preorder; Filtered set; Directed subset
In mathematics, a directed set (or a directed preorder or a filtered set) is a nonempty set A together with a reflexive and transitive binary relation \,\leq\, (that is, a preorder), with the additional property that every pair of elements has an upper bound.Kelley, p.