Intersection of Complete Meet Subsemilattices induces Closure Operator/Lemma

Lemma for Intersection of Complete Meet Subsemilattices induces Closure Operator

Let $\struct {S, \preccurlyeq}$ be an ordered set.

Let $\family {f_i}_{i \mathop \in I}$ be an indexed family of closure operators on $S$.

Let $C_i = \map {f_i} S$ be the set of closed elements with respect to $f_i$ for each $i \in I$.

Let $\family {C_i}_{i \mathop \in I}$ be an indexed family of complete meet subsemilattices of $S$.

Then $C = \ds \bigcap_{i \mathop \in I} C_i$ is also a complete meet subsemilattice.

Proof

Let $D \subseteq C$.

By Intersection is Largest Subset, $D \subseteq C_i$ for each $i \in I$.

Thus $D$ has an infimum in $S$ and $\inf D \in C_i$ for each $i \in I$.

By the definition of intersection, $\inf D \in C$.

$\blacksquare$