The intersection lattice of a hyperplane arrangement $A$ is the lattice of intersections in the arrangement partially ordered by containment.

i1 : R = RR[x,y]; |

i2 : A = {x + y, x, x - y, y + 1}; |

i3 : LA = intersectionLattice(A, R) -- warning: experimental computation over inexact field begun -- results not reliable (one warning given per session) o3 = LA o3 : Poset |

A theorem of Zaslavsky provides information about the topology of the complement of hyperplane arrangements over RR. In particular, the number of regions that $A$ divides RR into is derived from the moebiusFunction of the lattice. This can also be accessed with the realRegions method.

i4 : MF = moebiusFunction LA; |

i5 : sum apply(LA_*, i -> abs(MF#(ideal 0_R, i))) o5 = 10 |

Furthermore, the number of these bounded regions can also be extracted from the moebiusFunction of the lattice; see also boundedRegions.

i6 : MF' = moebiusFunction adjoinMax(LA, ideal 1_R); |

i7 : abs(MF'#(ideal 0_R, ideal 1_R)) o7 = 2 |

- boundedRegions -- computes the number of bounded regions a hyperplane arrangement divides the space into
- intersectionLattice -- generates the intersection lattice of a hyperplane arrangement
- moebiusFunction -- computes the Moebius function at every pair of elements of a poset
- realRegions -- computes the number of regions a hyperplane arrangement divides the space into