Definition:Lattice Ideal/Definition 2

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 $I$ is a join semilattice ideal

Also see

• Equivalence of Definitions of Lattice Ideal

Sources

• 1967: Saunders Mac Lane and Garrett Birkhoff: Algebra: Chapter $\text {XIV}$: Lattices, $\S 3$: Sublattices and Products of Lattices
• 1982: Peter T. Johnstone: Stone Spaces: Chapter $\text {I}$: Preliminaries, Definition $2.1$