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 $S$.

The following definitions of the concept of Lattice Ideal are equivalent:

Definition 1

$I$ is a lattice ideal of $S$ if and only if $I$ satisifes the lattice ideal axioms:

\((\text {LI 1})\)	$:$		$I$ is a sublattice of $S$:	\(\ds \forall x, y \in I:\)	\(\ds x \wedge y, x \vee y \in I \)
\((\text {LI 2})\)	$:$			\(\ds \forall x \in I: \forall a \in S:\)	\(\ds x \wedge a \in I \)

Definition 2

$I$ is a lattice ideal of $S$ if and only if $I$ is a join semilattice ideal

Proof

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:

$I$ is a lower section of $S$:

\(\ds \forall x \in F: \forall y \in S:\)

\(\ds y \preceq x \implies y \in I \)

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.

$\Box$

Definition 2 implies Definition 1

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

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

$\forall x \in I, a \in S: x \wedge a \in I$

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.

$\blacksquare$