Upper and lower sets
= Upper and lower sets
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In a <partially ordered set> $P$, an upper <set> $U$ satisfies $x\in U$ and $x\leq y\Rightarrow y\in U$; a lower <set> $L$ satisfies $x\in L$ and $y\leq x\Rightarrow y\in L$. The <complement of a set> exchanges the two conditions: if $x\notin U$ and $y\leq x$, 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.