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