Let $S$ be a finite set, and let $2^S$ be its powerset, regarded as a lattice. Let $L$ be a quotient (in the category of lattices and maps which preserve $\top,\bot,\wedge,\vee$) of $S$. What can we say about $L$?


 In fact, what I'd really like to know is: which finite semilattices are retracts (via $\bot,\vee$-preserving maps) of finite powerset lattices. I think the two questions are equivalent via a short-but-not-immediate argument.


Note that if $P$ is an arbitrary finite poset, then the lattice $2^P$ of poset maps $P \to 2$ is an example of such a lattice.



Which lattices are quotients of finite powerset lattices?
The class of finite powerset lattices is closed under quotients, up to isomorphism. That is, the quotients are exactly the lattice reducts of finite Boolean algebras.


In particular, any quotient $L$ of $2^S$ has to be a bounded distributive lattice, as the class of distributive lattices is a variety. Moreover, if $x\in L$ is an image of a set $A\subseteq S$, then the image of $S\smallsetminus A$ is an element $y$ such that $x\lor y=\top$ and $x\land y=\bot$. Thus, $L$ is complemented, i.e., it is a Boolean algebra, and as such it is isomorphic to $2^{S'}$ for some set $S'$.


