Definition:Dense (Lattice Theory)

Definition

Let $L = \struct {S, \wedge, \preceq}$ be a bounded below meet semilattice.

Dense Element

Let $x \in S$.

Then $x$ is dense if and only if

$\forall y \in S: y \ne \bot \implies x \wedge y \ne \bot$

where $\bot$ denotes the smallest element in $L$.

Dense Subset

Let $A$ be a subset of $S$.

Then $A$ is dense if and only if it includes only dense elements.

That means that if and only if $\forall x \in A: x$ is a dense element.