P A ( y ) := ∑ B ( − 1 ) | B | y dim f ( B ), any linear form defining the generic hyperplane of the arrangement. The characteristic polynomial of A, written p A( y), can be defined by 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.įor a subset B of A, let us define f( B) := the intersection of the hyperplanes in B this is S if B is empty. When L( A) is a lattice, the matroid of A, written M( A), has A for its ground set and has rank function r( S) := codim( I), where S is any subset of A and I is the intersection of the hyperplanes in S. (This is why the semilattice must be ordered by reverse inclusion-rather than by inclusion, which might seem more natural but would not yield a geometric (semi)lattice.) If the arrangement is linear or projective, or if the intersection of all hyperplanes is nonempty, the intersection lattice is a geometric lattice. The intersection semilattice L( A) is a meet semilattice and more specifically is a geometric semilattice. General theory The intersection semilattice and the matroid 1.1 The intersection semilattice and the matroid.
If S is 3-dimensional one has an arrangement of planes.
Historically, real arrangements of lines were the first arrangements investigated. If the whole space S is 2-dimensional, the hyperplanes are lines such an arrangement is often called an arrangement of lines. The intersection semilattice L( A) is partially ordered by reverse inclusion. These intersection subspaces of A are also called the flats of A. (excluding, in the affine case, the empty set). The intersection semilattice of A, written L( A), is the set of all 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. One may ask how these properties are related to the arrangement and its intersection semilattice. 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. In geometry and combinatorics, an arrangement of hyperplanes is an arrangement of a finite set A of hyperplanes in a linear, affine, or projective space S.
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