meet(Flat,Flat) -- compute the meet operation in the intersection lattice

Synopsis

Function: meet
Usage:

meet(F, G)

F ^ G
Inputs:
- F, a flat in an hyperplane arrangement,
- G, a flat in an hyperplane arrangement, in the same arrangement as $F$
Outputs:
- a flat in an hyperplane arrangement, having the greatest codimension among those contained in both $F$ and $G$

Description

In the geometric lattice of flats, the meet (also known as the infimum or greatest lower bound) is the intersection of the flats. Equivalently, identifying flats with subspaces, this operation is the Minkowski sum of the subspaces.

The meet operation is commutative, associative, and idempotent.

i1 : A = typeA 6;

i2 : F = flat(A, {0, 1, 6, 15, 20})

o2 = {0, 1, 6, 15, 20}

o2 : Flat of {x  - x , x  - x , x  - x , x  - x , x  - x , x  - x , x  - x , x  - x , x  - x , x  - x , x  - x , x  - x , x  - x , x  - x , x  - x , x  - x , x  - x , x  - x , x  - x , x  - x , x  - x }
               1    2   1    3   1    4   1    5   1    6   1    7   2    3   2    4   2    5   2    6   2    7   3    4   3    5   3    6   3    7   4    5   4    6   4    7   5    6   5    7   6    7

i3 : G = flat(A, {0, 1, 2, 6, 7, 11})

o3 = {0, 1, 2, 6, 7, 11}

o3 : Flat of {x  - x , x  - x , x  - x , x  - x , x  - x , x  - x , x  - x , x  - x , x  - x , x  - x , x  - x , x  - x , x  - x , x  - x , x  - x , x  - x , x  - x , x  - x , x  - x , x  - x , x  - x }
               1    2   1    3   1    4   1    5   1    6   1    7   2    3   2    4   2    5   2    6   2    7   3    4   3    5   3    6   3    7   4    5   4    6   4    7   5    6   5    7   6    7

i4 : H = flat(A, {0, 1, 2, 3, 6, 7, 8, 11, 12, 15})

o4 = {0, 1, 2, 3, 6, 7, 8, 11, 12, 15}

o4 : Flat of {x  - x , x  - x , x  - x , x  - x , x  - x , x  - x , x  - x , x  - x , x  - x , x  - x , x  - x , x  - x , x  - x , x  - x , x  - x , x  - x , x  - x , x  - x , x  - x , x  - x , x  - x }
               1    2   1    3   1    4   1    5   1    6   1    7   2    3   2    4   2    5   2    6   2    7   3    4   3    5   3    6   3    7   4    5   4    6   4    7   5    6   5    7   6    7

i5 : F ^ G

o5 = {0, 1, 6}

o5 : Flat of {x  - x , x  - x , x  - x , x  - x , x  - x , x  - x , x  - x , x  - x , x  - x , x  - x , x  - x , x  - x , x  - x , x  - x , x  - x , x  - x , x  - x , x  - x , x  - x , x  - x , x  - x }
               1    2   1    3   1    4   1    5   1    6   1    7   2    3   2    4   2    5   2    6   2    7   3    4   3    5   3    6   3    7   4    5   4    6   4    7   5    6   5    7   6    7

i6 : G ^ H

o6 = {0, 1, 2, 6, 7, 11}

o6 : Flat of {x  - x , x  - x , x  - x , x  - x , x  - x , x  - x , x  - x , x  - x , x  - x , x  - x , x  - x , x  - x , x  - x , x  - x , x  - x , x  - x , x  - x , x  - x , x  - x , x  - x , x  - x }
               1    2   1    3   1    4   1    5   1    6   1    7   2    3   2    4   2    5   2    6   2    7   3    4   3    5   3    6   3    7   4    5   4    6   4    7   5    6   5    7   6    7

i7 : F ^ H

o7 = {0, 1, 6, 15}

o7 : Flat of {x  - x , x  - x , x  - x , x  - x , x  - x , x  - x , x  - x , x  - x , x  - x , x  - x , x  - x , x  - x , x  - x , x  - x , x  - x , x  - x , x  - x , x  - x , x  - x , x  - x , x  - x }
               1    2   1    3   1    4   1    5   1    6   1    7   2    3   2    4   2    5   2    6   2    7   3    4   3    5   3    6   3    7   4    5   4    6   4    7   5    6   5    7   6    7

i8 : assert(meet(F, G) === F ^ G)

i9 : assert(F ^ G === G ^ F)

i10 : assert((F ^ G) ^ H === F ^ (G ^ H))

i11 : assert(G ^ G === G)

The rank function is also semimodular.

i12 : assert(rank F + rank G >= rank(F ^ G) + rank(F | G))

i13 : assert(rank F + rank H >= rank(F ^ H) + rank(F | H))

i14 : assert(rank H + rank G >= rank(H ^ G) + rank(H | G))

Ways to use this method: