Definition:Lattice Ideal

This page is about Ideal in the context of Lattice Theory. For other uses, see Ideal.

Definition

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

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

$I$ 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 $

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

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