Definition:Order Generating Subset

Definition

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

Let $X$ be a subset of $S$.

Then $X$ is order generating if and only if:

$\forall x \in S: x^\succeq \cap X$ admits an infimum and $x = \map \inf {x^\succeq \cap X}$

Also see

• Results about order generating subsets can be found here.

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_6:def 5