Inequality on the cardinal of a distributive lattice according to its height

Question

Let $L$ be a finite distributive lattice. Let $h(L)$ and $|L|$ be (respectively) the height and the cardinal of $L$.

Question: Is it true that $|L| \le 2^{h(L)}$ and that the equality holds iff $L$ is boolean?

Answer 1

Every distributive lattice can be embedded in a boolean lattice (this is a well-known fact about distributive lattices). Hence your inequality should hold with equality if $L$ is boolean.