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

## Synopsis

• Function: closure
• Usage:
closure(A,L) or closure(A,I)
• Inputs:
• A, , hyperplane arrangement
• L, a list, a list of indices of hyperplanes, or a linear ideal $I$ in the ring of ${\mathcal A}$
• Outputs:
• , 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 $I$.