Join Semilattice Ideal iff Ordered Set Ideal

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

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

Then:
 * $I$ is a join semilattice ideal of $\struct {S, \vee, \preceq}$ $I$ is an ordered set ideal of $\struct {S, \preceq}$.

Necessary Condition
Let $I$ be a join semilattice ideal of $\struct {S, \vee, \preceq}$.

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

Let $x, y \in I$.

Let $z = x \vee y$.

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

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

The result follows.

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

To show that $I$ is a join semilattice ideal of $\struct {S, \vee, \preceq}$ it is sufficient to show:

Let $x, y \in I$.

By definition of ordered set ideal, $I$ is a directed subset, so:
 * $\exists z \in I : x \preceq z \text{ and } y \preceq z$

By definition of join:
 * $x \vee y \preceq z$

By definition of ordered set ideal, $I$ is a lower section, so:
 * $x \vee y \in I$

The result follows.