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 $\struct {S, \vee, \preceq}$ if and only if $I$ is an ordered set ideal of $\struct {S, \preceq}$.

Proof

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:

$I$ is a directed subset of $S$:

\(\ds \forall x, y \in I: \exists z \in I:\)

\(\ds x \preceq z \text{ and } y \preceq z \)

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.

$\Box$

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:

\((\text {JSI 2})\)

$:$

$I$ is a subsemilattice of $S$:

\(\ds \forall x, y \in I:\)

\(\ds x \vee y \in I \)

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.

$\blacksquare$