The intersection lattice of a hyperplane arrangement $A$ is the lattice of intersections in the arrangement partially ordered by containment.
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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.
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Furthermore, the number of these bounded regions can also be extracted from the moebiusFunction of the lattice; see also boundedRegions.
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The source of this document is in /build/reproducible-path/macaulay2-1.25.05+ds/M2/Macaulay2/packages/Posets.m2:1899:0.