Editing Arrangement of hyperplanes (section)

== General theory ==

=== The intersection semilattice and the matroid ===

The intersection semilattice ''L''(''A'') is a meet semilattice and more specifically is a [[geometric semilattice]]. If the arrangement is linear or projective, or if the intersection of all hyperplanes is nonempty, the intersection lattice is a [[geometric lattice]].
(This is why the semilattice must be ordered by reverse inclusion&mdash;rather than by inclusion, which might seem more natural but would not yield a geometric (semi)lattice.)

When ''L''(''A'') is a lattice, the [[matroid]] of ''A'', written ''M''(''A''), has ''A'' for its ground set and has rank function ''r''(''B'') := codim(''f(B)''), where ''B'' is any subset of ''A'' and ''f(B)'' is the intersection of the hyperplanes in ''B''.  In general, when ''L''(''A'') is a semilattice, there is an analogous matroid-like structure called a [[semimatroid]], which is a generalization of a matroid (and has the same relationship to the intersection semilattice as does the matroid to the lattice in the lattice case), but is not a matroid if ''L''(''A'') is not a lattice.

=== Polynomials ===

For a subset ''B'' of ''A'', let us define ''f''(''B'') := the intersection of the hyperplanes in ''B''; this is ''S'' if ''B'' is empty.  
The '''characteristic polynomial of''' ''A'', written ''p<sub>A</sub>''(''y''), can be defined by 

:<math>p_A(y) := \sum_B (-1)^{|B|}y^{\dim f(B)},</math>

summed over all subsets ''B'' of ''A'' except, in the affine case, subsets whose intersection is empty.  (The dimension of the empty set is defined to be &minus;1.)  This polynomial helps to solve some basic questions; see below.
Another polynomial associated with ''A'' is the '''Whitney-number polynomial''' ''w<sub>A</sub>''(''x'', ''y''), defined by

:<math>w_A(x,y) := \sum_B x^{n-\dim f(B)} \sum_C (-1)^{|C-B|}y^{\dim f(C)},</math>

summed over ''B'' ⊆ ''C'' ⊆ ''A'' such that ''f''(''B'') is nonempty.

Being a geometric lattice or semilattice, ''L''(''A'') has a characteristic polynomial, ''p''<sub>''L''(''A'')</sub>(''y''), which has an extensive theory (see [[Matroid#Characteristic_polynomial|matroid]]).  Thus it is good to know that ''p''<sub>''A''</sub>(''y'') = ''y''<sup>''i''</sup> ''p''<sub>''L''(''A'')</sub>(''y''), where ''i'' is the smallest dimension of any flat, except that in the projective case it equals ''y''<sup>''i'' + 1</sup>''p''<sub>''L''(''A'')</sub>(''y'').  
The Whitney-number polynomial of ''A'' is similarly related to that of ''L''(''A'').  
(The empty set is excluded from the semilattice in the affine case specifically so that these relationships will be valid.)

=== The Orlik–Solomon algebra ===

The intersection semilattice determines another combinatorial invariant of the arrangement, the [[Orlik–Solomon algebra]]. To define it, fix a commutative subring ''K'' of the base field and form the [[exterior algebra]] ''E'' of the vector space
:<math>\bigoplus_{H \in A} K e_H </math>
generated by the hyperplanes.
A [[chain complex]] structure is defined on ''E'' with the usual boundary operator <math>\partial</math>.
The Orlik–Solomon algebra is then the quotient of ''E'' by the [[Ideal (ring theory)|ideal]] generated by elements of the form <math>e_{H_1} \wedge \cdots \wedge e_{H_p}</math> for which <math>H_1, \dots, H_p</math> have empty intersection, and by boundaries of elements of the same form for which <math>H_1 \cap \cdots \cap H_p</math> has [[codimension]] less than ''p''.