# intersection semilattice of a subspace arrangement

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

 $L(\mathcal{A})=\bigl{\{}\bigcap_{H\in\mathcal{S}}H\mid\mathcal{S}\subset% \mathcal{A}\bigr{\}}.$

Order (http://planetmath.org/Poset) the elements of $L(\mathcal{A})$ 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 $L(\mathcal{A})$. Moreover, the elements of $L(\mathcal{A})$ are naturally graded by codimension. If $\mathcal{A}$ happens to be a central arrangement, its intersection semilattice is in fact a lattice, with the meet operation defined by $H\wedge K=\operatorname{span}(H\cup K)$, where $\operatorname{span}(H\cup K)$ is the subspace of $V$ spanned by $H\cup K$.

Title intersection semilattice of a subspace arrangement IntersectionSemilatticeOfASubspaceArrangement 2013-03-22 15:47:58 2013-03-22 15:47:58 CWoo (3771) CWoo (3771) 8 CWoo (3771) Definition msc 52B99 msc 52C35 intersection lattice intersection semilattice