Meet Semilattice Filter iff Ordered Set Filter

Theorem
Let $\struct {S, \wedge, \preceq}$ be a meet semilattice.

Let $F \subseteq S$ be a non-empty subset of $S$.

Then:
 * $F$ is a meet semilattice filter of $\struct {S, \wedge, \preceq}$ $F$ is an ordered set filter of $\struct {S, \preceq}$.

Necessary Condition
Let $F$ be a meet semilattice ideal of $\struct {S, \wedge, \preceq}$.

To show that $F$ is an ordered set filter of $\struct {S, \preceq}$ it is sufficient to show:

Let $x, y \in F$.

Let $z = x \wedge y$.

By definition of meet semilattice ideal, $F$ is a subsemilattice, so:
 * $z \in F$

By definition of Meet:
 * $z \preceq x \text{ and } z \preceq y$

The result follows.

Sufficient Condition
Let $F$ be an ordered set filter of $\struct {S, \preceq}$.

To show that $F$ is a meet semilattice ideal of $\struct {S, \wedge, \preceq}$ it is sufficient to show:

Let $x, y \in F$.

By definition of ordered set filter:
 * $\exists z \in F : z \preceq x \text{ and } z \preceq y$

By definition of meet:
 * $z \preceq x \wedge y$

By definition of ordered set filter:
 * $x \wedge y \in F$

The result follows.