Definition:Lattice Ideal/Definition 1
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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 $S$: | \(\ds \forall x, y \in I:\) | \(\ds x \wedge y, x \vee y \in I \) | |||||
\((\text {LI 2})\) | $:$ | \(\ds \forall x \in I: \forall a \in S:\) | \(\ds x \wedge a \in I \) |
Also see
Sources
- 1971: George A. Grätzer: Lattice Theory: Chapter $1$: First Concepts, $\S 3$: Some Algebraic Concepts