# 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

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

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 $H\wedge K=\mathrm{span}(H\cup K)$, where
$\mathrm{span}(H\cup K)$ 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 |