 Upper and lower sets

In a partially ordered set $P$, an upper set $U$ satisfies $x\in U$ and $x\le y\Rightarrow y\in U$; a lower set $L$ satisfies $x\in L$ and $y\le x\Rightarrow y\in L$. The complement of a set exchanges the two conditions: if $x\notin U$ and $y\le x$, then $y\in U$ would force $x\in U$, so $P\setminus U$ is lower. In a Boolean lattice ordered by inclusion, these are the up-set and down-set conditions.