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 $S$ $I$ is a join semilattice ideal

Also see

 * Equivalence of Definitions of Lattice Ideal