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
Canonical name	IntersectionSemilatticeOfASubspaceArrangement
Date of creation	2013-03-22 15:47:58
Last modified on	2013-03-22 15:47:58
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	8
Author	CWoo (3771)
Entry type	Definition
Classification	msc 52B99
Classification	msc 52C35
Synonym	intersection lattice
Synonym	intersection semilattice