closure(Arrangement,List) -- closure operation in the intersection lattice

Synopsis

Function: closure
Usage:

closure(A,L) or closure(A,I)
Inputs:
- A, a hyperplane arrangement, hyperplane arrangement
- L, a list, a list of indices of hyperplanes, or a linear ideal $I$ in the ring of ${\mathcal A}$
Outputs:
- a flat in an hyperplane arrangement, the flat of least codimension containing the hyperplanes $L$, or the flat consisting of those hyperplanes of $\mathcal A$ whose defining forms are also in $I$

Description

The closure of a set of indices $L$ consists of (indices of) all hyperplanes that contain the intersection of the given ones.

Equivalently, the closure of $L$ consists of all hyperplanes whose defining linear forms are in the span of the linear forms indexed by $L$.

i1 : A = typeA 3

o1 = {x  - x , x  - x , x  - x , x  - x , x  - x , x  - x }
       1    2   1    3   1    4   2    3   2    4   3    4

o1 : Hyperplane Arrangement

i2 : F = closure(A,{0,1})

o2 = {0, 1, 3}

o2 : Flat of {x  - x , x  - x , x  - x , x  - x , x  - x , x  - x }
               1    2   1    3   1    4   2    3   2    4   3    4

i3 : A_F

o3 = {x  - x , x  - x , x  - x }
       1    2   1    3   2    3

o3 : Hyperplane Arrangement

i4 : I = ideal((hyperplanes A)_{0,3}) -- one can also specify a linear ideal

o4 = ideal (x  - x , x  - x )
             1    2   2    3

o4 : Ideal of QQ[x ..x ]
                  1   4

i5 : assert (F == closure(A,I))

The closure of a linear ideal $I$ is the flat consisting of all the hyperplanes in $\mathcal A$ whose defining forms are also in $I$.

Ways to use this method: