The subspaces of some n+1 dimensional vector space V form a lattice where lattice ‘meet’ (x⋀y) of subspaces x and y is subspace intersection and lattice ‘join’ (x⋁y) is the span of the union of the subspaces. The lattice ordering, ≤, corresponds to subspace inclusion, ⊆. Also x≤y iff x = x⋀y. Subspaces form a bounded lattice with the whole space V serving as the lattice 1, and subspace {0} serving as the lattice 0.

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.