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 $\struct {S, \wedge, \preceq}$ if and only if $F$ is an ordered set filter of $\struct {S, \preceq}$.

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

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

$\ds \forall x, y \in F: \exists z \in F:$

$\ds z \preceq x \text{ and } z \preceq y $

Let $x, y \in F$.

Let $z = x \wedge y$.

$z \in F$

By definition of Meet:

$z \preceq x \text{ and } z \preceq y$

The result follows.

$\Box$

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

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

$F$ is a subsemilattice of $S$:

$\ds \forall x, y \in F:$

$\ds x \wedge y \in F $

Let $x, y \in F$.

$\exists z \in F : z \preceq x \text { and } z \preceq y$

By definition of meet:

$z \preceq x \wedge y$

$x \wedge y \in F$

The result follows.

$\blacksquare$