Structure of the set of relations with $\cup$ and (maybe) $\cap$

Question

Let $p_1,p_2 \text{ and } q$ be relations. Then it is easy to show that composition of relations distributes over union. That is, $$(p_1 \cup p_2) \circ q = (p_1 \circ q) \cup (p_2 \cup q)$$ What about intersection? Does it distribute in some way? Also, should I be aware of some structure over the set of relations with $\cup$ operation? Maybe a lattice? To give you some context this is for a course on software verification were programs are defined in terms of relations between input and output variables.