# Ideals form Complete Lattice

## Theorem

Let $L = \struct {S, \vee, \preceq}$ be a bounded below join semilattice.

Let $\mathcal I = \struct {\mathit{Ids}\left({L}\right), \subseteq}$ be an inclusion ordered set,

where $\mathit{Ids}\left({L}\right)$ denotes the set of all ideals in $L$.

Then $\mathcal I$ is complete lattice.

## Proof

Let $X \subseteq \mathit{Ids}\left({L}\right)$.

$\bigcap X$ is an ideal.
$\forall A \in \mathit{Ids}\left({L}\right):\left({\forall I \in X: A \subseteq I}\right) \iff A \subseteq \bigcap X$

Thus by definition:

$X$ admits an infimum.

Thus by dual to Lattice is Complete iff it Admits All Suprema:

$\mathcal I$ is complete lattice.

$\blacksquare$