Atomistic lattice

Remarks

The least element $0$ is the join of the empty set of atoms. An atomistic lattice is atomic: if $b \ne 0$ had no atom below it, $b$ would be the join of the empty set, that is $b = 0$. Example: the power set of a set $X$ is atomistic, since every subset $B$ is the union of the singletons $\set{x}$ with $x \in B$. Non-example: the chain $0 < 1 < 2$ is atomic, with the single atom $1$, but not atomistic. The join of the atoms below $2$ is $1 \ne 2$.