By lattice theory, meet and join each commute and associate. They also absorp: x⋁(x⋀y) = x and x⋀(x⋁y) = x, and are idempotent: x⋁x = x and x⋀x = x. Notice the duality between meet and join. Every axiom and thus every theorem in these operators remain true when ⋁ and ⋀ are exchanged. This mirrors duality in vector spaces. Some lattices and in particular those of subspaces are symmetric top to bottom. Not only are the theorems symmetric, but the lattices too; there are one to one maps from top to bottom.