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# intersection semilattice of a subspace arrangement

Let $\mathcal{A}$ be a finite subspace arrangement in a finite-dimensional vector space $V$. The intersection semilattice of $\mathcal{A}$ is the subspace arrangement $L(\mathcal{A})$ defined by taking the closure 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 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 $H\wedge K=\Span(H\cup K)$, where $\Span(H\cup K)$ is the subspace of $V$ spanned by $H\cup K$.

## Mathematics Subject Classification

52B99*no label found*52C35

*no label found*

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