Definition:Meet Semilattice Filter

Definition

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

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

Then $F$ is a meet semilattice ideal of $S$ if and only if $F$ satisifies the meet semilattice filter axioms:

$(\text {MSF 1})$	$:$		$F$ is an upper section of $S$:	$\ds \forall x \in F: \forall y \in S:$	$\ds x \preccurlyeq y \implies y \in F $
$(\text {MSF 2})$	$:$		$F$ is a subsemilattice of $S$:	$\ds \forall x, y \in F:$	$\ds x \wedge y \in F $

1982: Peter T. Johnstone: Stone Spaces: Chapter $\text {I}$: Preliminaries, Definition $2.2$

\((\text {MSF 1})\)	$:$		$F$ is an upper section of $S$:	\(\ds \forall x \in F: \forall y \in S:\)	\(\ds x \preccurlyeq y \implies y \in F \)
\((\text {MSF 2})\)	$:$		$F$ is a subsemilattice of $S$:	\(\ds \forall x, y \in F:\)	\(\ds x \wedge y \in F \)