Separative forcing order ($p\nleqq q⟹\exists r\le p(r\perp q)$)

In standard notation for forcing, an order is separative if $p\nleqq q$ implies that some $r\le p$ is incompatible with $q$. Separativity does not imply an atomless forcing order: the one-point order is separative and has a forcing atom. Its only generic filter is a ground-model set. Atomlessness is the hypothesis in generic filter for an atomless order is new.