Axiom:Join Semilattice Ideal Axioms

Definition

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

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

$I$ is a join semilattice ideal of $S$ if and only if $I$ satisifes the axioms:

\((\text {JSI 1})\)	$:$		$I$ is a lower section of $S$:	\(\ds \forall x \in I: \forall y \in S:\)	\(\ds y \preceq x \implies y \in I \)
\((\text {JSI 2})\)	$:$		$I$ is a subsemilattice of $S$:	\(\ds \forall x, y \in I:\)	\(\ds x \vee y \in I \)

These criteria are called the join semilattice ideal axioms.