Editing Arrangement of hyperplanes (section)

{{short description|Partition of space by a hyperplanes}}
In [[geometry]] and [[combinatorics]], an '''arrangement of hyperplanes''' is an [[arrangement (space partition)|arrangement]] of a finite set ''A'' of [[hyperplane]]s in a [[linear space|linear]], [[affine geometry|affine]], or [[projective geometry|projective]] space ''S''.  
Questions about a hyperplane arrangement ''A'' generally concern geometrical, topological, or other properties of the '''complement''', ''M''(''A''), which is the set that remains when the hyperplanes are removed from the whole space.  One may ask how these properties are related to the arrangement and its intersection semilattice.
The '''intersection [[semilattice]]''' of ''A'', written ''L''(''A''), is the set of all [[Euclidean subspace|subspaces]] that are obtained by intersecting some of the hyperplanes; among these subspaces are ''S'' itself, all the individual hyperplanes, all intersections of pairs of hyperplanes, etc. (excluding, in the affine case, the empty set).  These '''intersection subspaces''' of ''A'' are also called the '''flats of''' ''A''.  The intersection semilattice ''L''(''A'') is partially ordered by ''reverse inclusion''.  

If the whole space ''S'' is 2-dimensional, the hyperplanes are [[line (mathematics)|line]]s; such an arrangement is often called an '''[[arrangement of lines]]'''.  Historically, real arrangements of lines were the first arrangements investigated.  If ''S'' is 3-dimensional one has an '''arrangement of planes'''.
[[File:Arrangement hyperplans.png|thumbnail|A hyperplane arrangement in space]]