Do all equational theorems of Boolean algebra not involving complementation also hold for all bounded distributive lattices?

Question

Or we might ask the question in the negative:

Do there exist equational theorems of Boolean algebra involving only the operations $\wedge,\vee$ and the constants $\top$ and $\bot$ that fail to be theorems of bounded distributive lattice?

Answer 1

All equations hold. This is because we have the following result:

Proposition. For each distributive lattice $L$, there exists a set $X$ and an embedding $i : L \to \mathscr{P}(X)$ that preserves finite meets and joins.

This is a corollary of Stone's representation theorem for distributive lattices:

Theorem. For each distributive lattice $L$, there exists a (unique up to homeomorphism) topological space $\operatorname{Spec} L$ with the following properties:

• $\operatorname{Spec} L$ is quasicompact and sober.
• The set of quasicompact open subsets of $\operatorname{Spec} L$ is closed under finite intersections and is a basis for the topology of $\operatorname{Spec} L$.
• The lattice of quasicompact open subsets of $\operatorname{Spec} L$ is isomorphic to $L$.

In particular, there is a lattice embedding $L \to \mathscr{P}(\operatorname{Spec} L)$.