Equivalence of Definitions of Lattice Ideal

Theorem
Let $\struct {S, \vee, \wedge, \preceq}$ be a lattice.

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

Definition 1 implies Definition 2
Let $I$ satisify the lattice ideal axioms.

To show that $I$ is a join semilattice ideal it is sufficient to show:

Let $x \in I, y \in S : y \preceq x$.

By the lattice ideal axioms, $I$ is a sublattice of $S$, so:
 * $x \wedge y \in I$

By definition of sublattice and lattice:
 * $\struct{I, \wedge}$ is a meet semilattice

From Preceding iff Meet equals Less Operand:
 * $y = x \wedge y$

Hence:
 * $y \in I$

The result follows.

Definition 2 implies Definition 1
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.