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, $F$ is a sublattice of $\struct {S, \vee, \wedge, \preceq}$, so:
 * $x \wedge y \in I$

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 a join semilattice filter of $\struct {S, \vee, \preceq}$.

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

Let $x \in I, a \in S$.

By definition of meet:
 * $x \wedge a \preceq x$

By definition of join semilattice ideal, $I$ is an lower section, so:
 * $x \wedge a \in I$

The result follows.