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Posets :: Example: Intersection lattices

Example: Intersection lattices

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

See also