Editing Arrangement of hyperplanes (section)

=== 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.