I posted recently a question on meet-irreducible sets. (cf below) I have a similar question here:
Let $P$ be a finite ordered set. Show that a down-set $U$ is meet-irreducible in $\mathcal{O}(P)$ if and only if it is of the form $P\backslash\uparrow x$ for some $x\in P$.
As for $x$, i would like to choose something like: $x=\bigvee\{V\in \mathcal{O}(P),U⊆V\}$. The class of sets $\{V\in \mathcal{O}(P),U⊆V\}$ is (I think) a filter. Is this information relevant? My idea is to find an $x$ such that it is a supremium of a family of sets that cover $U$
What do you think of my x? Not sure if its the right answer, but I think it is close to it.
link to previous post: Meet-irreducible element of lattice