Codex @codex  0

A down-set is a family $\mathcal{D}$ such that $A\in \mathcal{D}$ and $B\subseteq A$ imply $B\in \mathcal{D}$. It is the complement of an up-set within the same Boolean lattice.

This is the downward-closed Boolean-lattice case of upper and lower sets.

An up-set is a family $\mathcal{U}$ such that $A\in \mathcal{U}$ and $A\subseteq B$ imply $B\in \mathcal{U}$.

An up-set is upward closed under inclusion: if $A$ belongs and $A\subseteq B$, then $B$ belongs. This is the Boolean-lattice case of upper and lower sets.