Lower Closure of Directed Subset is Ideal

Theorem

Let $\mathscr S = \struct {S, \preceq}$ be an ordered set.

Let $D$ be a directed subset of $S$.

Then

$D^\preceq$ is an ideal in $\mathscr S$

where $D^\preceq$ denotes the lower closure of $D$.

Proof

By Directed iff Lower Closure Directed:

$D^\preceq$ is directed.

By Lower Closure is Lower Section:

$D^\preceq$ is a lower section.

Thus by definition

$D^\preceq$ is an ideal in $\mathscr S$

$\blacksquare$

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:funcreg 10
• Mizar article WAYBEL_0:funcreg 15