Intersection of Semilattice Ideals is Ideal/Set of Sets

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Theorem

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

Let $\II$ be a set of ideals in $\struct {S, \preceq}$.


Then $\bigcap \II$ is an ideal in $\struct {S, \preceq}$.


Proof

Non-Empty Set

By Bottom in Ideal:

$\forall I \in \II: \bot \in I$

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

By definition of intersection:

$\bot \in \bigcap \II$

Hence $\bigcap \II$ is non-empty.

$\Box$


Lower Set

Let $x \in \bigcap \II$, $y \in S$ such that

$y \preceq x$

We will prove that:

$\forall I \in \II: y \in I$

Let $I \in \II$.

By definition of intersection:

$x \in I$

Thus by definition of lower set:

$y \in I$

$\Box$

Thus by definition of intersection:

$y \in \bigcap \II$

$\Box$


Directed Subset

Let $x, y \in \bigcap \II$.

We will prove that:

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

Let $I \in \II$.

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:

$x \vee y = \sup \set {x, y}$

By definition:

$z$ is an upper bound for $\set {x, y}$.

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:

$x \vee y \in \bigcap \II$

Hence by Directed in Join Semilattice:

$\bigcap \II$ is directed.

$\blacksquare$


Sources