Are all set lattices isomorphic to a power set? A set lattice is a lattice where the meet is the intersection, the join is the union, and the partial order is $\subseteq$. The standard example is is $P(A)$ where $A$ is a set. I’ve thought about other possible examples, yet have failed to come up with any. Are there any set/distributive lattices that aren’t isomorphic to $P(A)$ for some $A$?
 A: Well, not every lattice is isomorphic to $P(A)$. For example the $P(A)$ lattice has $2^{|A|}$ elements, and there are lattices for any order, e.g.
$$K_n=\big\{\{1\},\{1,2\},\ldots,\{1,2,\ldots,n-1,n\}\big\}$$
with union and intersection is a latice of exactly $n$ elements.

So the more interesting question would be: can every lattice be embedded into $P(A)$ for some set $A$? The answer is still no. To see that we need to look at some property of (a sublattice of) $P(A)$ that other lattices don't have to have. One of those properties is the distributivity of $\vee$ over $\wedge$ (and vice versa). Union is always distributive over intersection (and vice versa), and so any subtlatice of $P(A)$ is distributive. But there are non-distributive lattices, e.g the $M_3$:
$$M_3=\{0,a,b,c,1\}$$
$$0< a<1$$
$$0< b<1$$
$$0< c<1$$
In fact this is if and only if: a lattice is isomorphic to a lattice of sets (with union and intersection), and thus embeddable into some $P(A)$, if and only if it is distributive. Note that this is a non-trivial theorem (see the wiki I linked earlier).
A: Total orders with $\land = \min$ and $\lor = \max$ are lattices and any of them, call it $(X,\le)$ and call $(-\infty,x)=\{y\in X\,:\, y<x\}$, is isomorphic to the set lattice $(\{(-\infty,x)\,:\, x\in X\}, \subseteq)$. However, $P(A)$ is hardly ever totally ordered.
