Intersection of Semilattice Ideals is Ideal/Set of Sets

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Theorem

Let $\left({S, \preceq}\right)$ be a bounded below join semilattice.

Let $\mathcal I$ be a set of ideals in $\left({S, \preceq}\right)$.


Then $\bigcap \mathcal I$ is an ideal in $\left({S, \preceq}\right)$.


Proof

Non-Empty Set

By Bottom in Ideal:

$\forall I \in \mathcal I: \bot \in I$

where $\bot$ denotes the smallest element in $S$.

By definition of intersection:

$\bot \in \bigcap \mathcal I$

Hence $\bigcap \mathcal I$ is non-empty.

$\Box$

Lower Set

Let $x \in \bigcap \mathcal I$, $y \in S$ such that

$y \preceq x$

We will prove that

$\forall I \in \mathcal I: y \in I$

Let $I \in \mathcal I$.

By definition of intersection:

$x \in I$

Thus by definition of lower set:

$y \in I$

$\Box$

Thus by definition of intersection:

$y \in \bigcap \mathcal I$

$\Box$


Directed Subset

Let $x, y \in \bigcap \mathcal I$.

We will prove that

$\forall I \in \mathcal I: x \vee y \in I$

Let $I \in \mathcal I$.

By definition of intersection:

$x, y \in I$

By definition of directed:

$\exists z \in I: x \preceq z \land y \preceq z$

By definition of join:

$x \vee y = \sup \left\{ {x, y}\right\}$

By definition:

$z$ is upper bound for $\left\{ {x, y}\right\}$.

By definition of supremum:

$x \vee y \preceq z$

Thus by definition of lower set:

$x \vee y \in I$

$\Box$

Thus by definition of intersection:

$x \vee y \in \bigcap \mathcal I$

Hence by Directed in Join Semilattice:

$\bigcap \mathcal I$ is directed.

$\blacksquare$


Sources