intersection semilattice of a subspace arrangement

Let $𝒜$ be a finite subspace arrangement in afinite-dimensional vector space $V$.The of$𝒜$ is the subspacearrangement $L(𝒜)$ defined by taking theclosure (http://planetmath.org/ClosureAxioms)of $𝒜$ under intersections. More formally, let

$$L(𝒜)=\{\bigcap _{H\in 𝒮}H\mid 𝒮\subset 𝒜\}.$$

Order (http://planetmath.org/Poset) the elements of $L(𝒜)$by reverse inclusion,and give it the structure of a join-semilattice by defining$H\vee K=H\cap K$ for all $H$, $K$ in $L(𝒜)$.Moreover, the elements of $L(𝒜)$ are naturallygraded by codimension. If $𝒜$happens to be a central arrangement, its intersectionsemilattice is in fact a lattice, with the meet operationdefined by $H\wedge K=\mathrm{span}(H\cup K)$, where$\mathrm{span}(H\cup K)$ is the subspace of $V$ spanned by$H\cup K$.