# Lower Closure of Directed Subset is Ideal

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## Theorem

Let $\mathscr S = \left({S, \preceq}\right)$ 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 Set:

- $D^\preceq$ is a lower set.

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