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 $S$ $I$ satisifes the axioms:

These criteria are called the join semilattice ideal axioms.