Definition:Directed Subset

Definition

Let $\struct {S, \precsim}$ be a preordered set.

Let $H$ be a non-empty subset of $S$.

Then $H$ is a directed subset of $S$ if and only if:

$\forall x, y \in H: \exists z \in H: x \precsim z$ and $y \precsim z$

Also see

• Definition:Directed Set

Sources

• 1980: G. Gierz, K.H. Hofmann, K. Keimel, J.D. Lawson, M.W. Mislove and D.S. Scott: A Compendium of Continuous Lattices

• Mizar article WAYBEL_0:def 1